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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf,len=2106
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='02848v1  [math-ph]  7 Jan 2023  BOUNDEDNESS OF THE FIFTH OFF-DIAGONAL DERIVATIVE FOR  THE ONE-PARTICLE COULOMBIC DENSITY MATRIX  PETER HEARNSHAW  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Boundedness is demonstrated for the fifth derivative of the one-particle  reduced density matrix for non-relativistic Coulombic wavefunctions in the vicinity of  the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Introduction and results  We consider the non-relativistic quantum systems of N ≥ 2 electrons among N0 nuclei  which represents the system of an atom or molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For simplicity we restrict ourselves  to the case of an atom (N0 = 1), although all results readily generalise to the molecular  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The electrons have coordinates x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xN), xk ∈ R3, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N, and  the nucleus has charge Z > 0 and its position fixed at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The corresponding  Schr¨odinger operator is  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1)  H = −∆ + V  where ∆ = �N  k=1 ∆xk is the Laplacian in R3N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ∆xk refers to the Laplacian applied  to the variable xk, and V is the Coulomb potential given by  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2)  V (x) = −  N  �  k=1  Z  |xk| +  �  1≤j<k≤N  1  |xj − xk|  for x ∈ R3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This operator acts in L2(R3N) and is self-adjoint on H2(R3N), see for  example [1, Theorem X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We consider solutions to the eigenvalue problem for H in  the operator sense, namely  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3)  Hψ = Eψ  for ψ ∈ H2(R3N) and E ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N, we represent  ˆxj = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xj−1, xj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xN),  (x, ˆxj) = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xj−1, x, xj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xN)  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Primary 35B65;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Secondary 35J10, 81V55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Multi-particle quantum system, Coulombic wavefunction, Schr¨odinger equa-  tion, one-particle density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  1  2  PETER HEARNSHAW  for x ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We define the one-particle reduced density matrix, or simply density matrix,  by  γ(x, y) =  �  R3N−3 ψ(x, ˆx)ψ(y, ˆx) dˆx,  ˆx = ˆx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4)  for x, y ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' More commonly, the one-particle reduced density matrix is defined as the  function  g(x, y) =  N  �  j=1  �  R3N−3  ψ(x, ˆxj)ψ(y, ˆxj) dˆxj,  x, y ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5)  However, since we are interested only in regularity properties we need only study one  term of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5), hence the definition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In fact we have g(x, y) = Nγ(x, y) whenever  ψ is totally antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' An important related function is the one-particle density, or  simply density, which is defined here as  ρ(x) = γ(x, x) =  �  R3N−3 |ψ(x, ˆx)|2 dˆx,  x ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6)  Now suppose ψ is any eigenfunction obeying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3) with ∥ψ∥L2(R3N) = 1, and let ρ and  γ be the corresponding functions as defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In [2], real  analyticity is proven for γ(x, y) as a function of two variables in the set  D = {(x, y) ∈ R3 × R3 : x ̸= 0, y ̸= 0, x ̸= y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  In particular, real analyticity was not shown across the diagonal, that is where x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Previously, real analyticity was shown to hold for the density ρ(x) on the set R3\\{0} in  [3], see also [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In other words, this shows real analyticity of γ in one variable along the  diagonal x = y, excluding the point x = y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Despite this, there is no smoothness of  γ across the diagonal which is discussed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Pointwise derivative estimates for γ on  the set D were then given in this paper, namely [5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1], and are restated as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For b ≥ 0, t > 0 define  hb(t) =  �  tmin{0,5−b}  if b ̸= 5  log(t−1 + 2)  if b = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We denote N0 = N ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For all R > 0 and α, β ∈ N3  0 with |α|, |β| ≥ 1 there exists C  such that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7)  |∂α  x ∂β  y γ(x, y)| ≤ C  �  1 + |x|2−|α|−|β| + |y|2−|α|−|β|  + h|α|+|β|(|x − y|)  �  ∥ρ∥1/2  L1(B(x,R)) ∥ρ∥1/2  L1(B(y,R))  and for all |α| ≥ 1 there exists C such that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8)  |∂α  x γ(x, y)| + |∂α  y γ(x, y)| ≤ C  �  1 + |x|1−|α| + |y|1−|α|  + h|α|(|x − y|)  �  ∥ρ∥1/2  L1(B(x,R)) ∥ρ∥1/2  L1(B(y,R))  ONE-PARTICLE DENSITY MATRIX  3  for all x, y ∈ R3 with x ̸= 0, y ̸= 0 and x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The notation ∂α  x refers to the α-partial  derivative in the x variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The constant C depends on α, β, R, N and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The right-  hand side is finite because ψ is normalised and hence ρ ∈ L1(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The bounded first  derivative at the nucleus reflects local Lipschitz continuity of γ on R6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In addition, there  is local boundedness of up to four derivatives at the diagonal with at worst a logarithmic  singularity for the fifth derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The purpose of this current paper is to show local  boundedness of the fifth derivative at the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The existence of the fifth-order cusp at the diagonal was previously demonstrated in  [6] (see also [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In this paper, quantum chemistry calculations show that for x, r ∈ R3,  x ̸= 0 and small r we have  Re[γ(x + r, x − r)] = γ(x, x) + C(x)|r|5 + R(x, r)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9)  for some functions C(x) and R(x, r), the latter having no contribution from |r|k for  k = 0, 1, 3, 5 in the small |r| expansion at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Our main result is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let ψ be an eigenfunction of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Define m(x, y) = min{1, |x|, |y|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Then for all |α| + |β| = 5 and R > 0 we have C, depending on R and also on Z, N and  E, such that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10)  |∂α  x ∂β  y γ(x, y)| ≤ Cm(x, y)−4 ∥ρ∥1/2  L1(B(x,R)) ∥ρ∥1/2  L1(B(y,R))  for all x, y ∈ R3 obeying 0 < |x − y| ≤ (2N)−1m(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (1) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1 naturally extends to the case of a molecule with  several nuclei whose positions are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The modifications are straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (2) The bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10) naturally complements (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8) for |α| + |β| ̸= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (3) As a consequence of [5, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1], the inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10) shows that γ ∈  C4,1  loc  �  (R3\\{0}) × (R3\\{0})  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' As mentioned earlier, we use standard notation whereby x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xN) ∈  R3N, xj ∈ R3, j=1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N, and where N is the number of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In addition, define  for 1 ≤ j, k ≤ N, j ̸= k,  ˆxj = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xj−1, xj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xN)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='11)  ˆxjk = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xj−1, xj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xN)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='12)  with obvious modifications if either j, k equals 1 or N, and if k < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We define ˆx = ˆx1,  which will be used throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Variables placed before ˆxj and ˆxjk will be placed in the  removed slots as follows, for any x, y ∈ R3 we have  (x, ˆxj) = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xj−1, x, xj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xN),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='13)  (x, y, ˆxjk) = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xj−1, x, xj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xk−1, y, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14)  In this way, x = (xj, ˆxj) = (xj, xk, ˆxjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  4  PETER HEARNSHAW  We define a cluster to be any subset P ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Denote P c = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N}\\P,  P ∗ = P\\{1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We will also need cluster sets, P = (P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , PM), where M ≥ 1 and  P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , PM are clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  First-order cluster derivatives are defined, for a non-empty cluster P, by  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='15)  Dα  P =  �  j∈P  ∂α  xj  for α ∈ N3  0, |α| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  For P = ∅, Dα  P is defined as the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Higher order cluster derivatives, for α =  (α′, α′′, α′′′) ∈ N3  0 with |α| ≥ 2, are defined by successive application of first-order cluster  derivatives as follows,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16)  Dα  P = (De1  P )α′(De2  P )α′′(De3  P )α′′′  where e1, e2, e3 are the standard unit basis vectors of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let P = (P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , PM) and  α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , αM), αj ∈ N3  0, 1 ≤ j ≤ M, then we define the multicluster derivative (often  simply referred to as cluster derivative) by  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='17)  Dα  P = Dα1  P1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' DαM  PM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  It can readily be seen that cluster derivatives obey the Leibniz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Throughout, the letter C refers to a positive constant whose value is unimportant but  may depend on Z, N and the eigenvalue E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Distance function notation and elementary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For non-empty cluster P,  define  ΣP =  �  x ∈ R3N :  �  j∈P  |xj|  �  l∈P  m∈P c  |xl − xm| = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='18)  For P = ∅ we set ΣP := ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Denote Σc  P = R3N\\ΣP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For each P we have ΣP ⊂ Σ where  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='19)  Σ =  �  x ∈ R3N :  �  1≤j≤N  |xj|  �  1≤l<m≤N  |xl − xm| = 0  �  is the set of singularities of the Coulomb potential V , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For any cluster P we can  define the following two distances  dP(x) := dist(x, ΣP) = min  �  |xj|, 2−1/2|xj − xk| : j ∈ P, k ∈ P c�  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20)  λP(x) := min{1, dP(x)}  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='21)  for all x ∈ R3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Using the formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20), see for example [5, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2], it can be shown that both  dP and λP are Lipschitz and obey  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='22)  |dP(x) − dP(y)|, |λP(x) − λP(y)| ≤ |x − y|  for all x, y ∈ R3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ONE-PARTICLE DENSITY MATRIX  5  Let P = (P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , PM) be a cluster set and α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , αM) ∈ N3M  0  be a multiindex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Define  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='23)  Σα =  �  j : αj̸=0  ΣPj  for non-zero α and when α = 0 we set Σα = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Denote Σc  α = R3N\\Σα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For non-zero α  we can also define the two distances  dα(x) = min{dPj(x) : αj ̸= 0, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , M},  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='24)  λα(x) = min{λPj(x) : αj ̸= 0, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , M}  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='25)  for all x ∈ R3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Notice we also have the identity  dα(x) = dist(x, Σα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Indeed, since ΣPj ⊂ Σα whenever j is such that αj ̸= 0, we have dist(x, Σα) ≤ dPj(x)  and hence dist(x, Σα) ≤ dα(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Conversely for each ξ ∈ Σα we have ξ ∈ ΣPj for some j  with αj ̸= 0, and hence |x − ξ| ≥ dPj(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore, |x − ξ| ≥ dα(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Taking infimum  over ξ ∈ Σα we obtain the reverse inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We will also use the following related quantities involving maxima of relevant distances  for non-zero α,  qα(x) = max{dPj(x) : αj ̸= 0, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , M}  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='26)  µα(x) = max{λPj(x) : αj ̸= 0, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='27)  Finally, for α = 0 we set dα, qα ≡ 0 and λα, µα ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  In order to state the results we first define the following for an arbitrary function u  and any r > 0,  f∞(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' u) := ∥∇u∥L∞(B(x,r)) + ∥u∥L∞(B(x,r))  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='28)  for x ∈ R3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The ball B(x, r) is considered in R3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Largely, this notation will be used  for u = ψ and in this case we have the notation,  f∞(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' r) := f∞(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='29)  A pointwise cluster derivative bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1 we will state and  prove a new pointwise bound to cluster derivatives of eigenfunctions ψ, which may itself  be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' It will be shown by elliptic regularity that for all α the weak  cluster derivatives Dα  Pψ exist in the set Σc  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' It is therefore interesting to consider how  such cluster derivatives behave as the set Σα is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Previously, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Fournais and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Ø.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Sørensen have given bounds to local Lp-norms of  cluster derivatives of ψ for a single cluster P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Indeed, in [8, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10] it is shown  that for any multiindex 0 ̸= α ∈ N3  0, p ∈ (1, ∞] and any 0 < r < R < 1 there exists C,  depending on r, R, p and α, such that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='30)  ∥Dα  Pψ∥Lp(B(x,rλP (x))) ≤ CλP(x)1−|α|�  ∥∇ψ∥Lp(B(x,RλP (x))) + ∥ψ∥Lp(B(x,RλP (x)))  �  6  PETER HEARNSHAW  for all x ∈ Σc  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Notice that for every x ∈ Σc  P, we have B(x, rλP(x)) ⊂ Σc  P by the  definition of λP(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore, the set ΣP is avoided when evaluating Dα  Pψ in the Lp-  norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The objective of the following theorem is to extend the bounds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='30) in the case of  p = ∞ and for cluster derivatives for cluster sets P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In particular, estimates are obtained  which depend on the order of derivative for each of the respective clusters in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In the  following, ∇ denotes the gradient operator in R3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For every cluster set P = (P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , PM), multiindex α ∈ N3M  0  and any  0 < r < R < 1 there exists C, depending on α, r and R, such that for k = 0, 1,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='31)  ��Dα  P∇kψ  ��  L∞(B(x,rλα(x))) ≤ Cλα(x)1−kλP1(x)−|α1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|αM|f∞(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)  for all x ∈ Σc  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Furthermore, for each |α| ≥ 1 there exists a function Gα  P : Σc  α → C3N such that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='32)  Dα  P∇ψ = Gα  P + ψDα  P∇Fc  and for every b ∈ [0, 1) there exists C, depending on α, r, R and b, such that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='33)  ∥Gα  P∥L∞(B(x,rλα(x))) ≤ Cµα(x)bλP1(x)−|α1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|αM|f∞(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)  for all x ∈ Σc  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (1) In the case of a single cluster and k = 0, the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='31) reestab-  lishes (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='30) in the case of p = ∞, albeit with a slightly larger radius in the  L∞-norms on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (2) When k = 0 and α ̸= 0, the presence of a single power of λα(x) in the bound  will cancel a single negative power of λPj(x) for j such that λPj(x) ≤ λPi(x) for  all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , M with αi ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Notice that the appropriate j will depend on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (3) The bound in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='33) is stronger than that of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='31) with k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This is because  a positive power of µα(x) will partially cancel a single negative power of λPj(x)  for j such that λPj(x) ≥ λPi(x) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , M with αi ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3 will follow a similar strategy to that of [8, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  An additional result will be required to prove (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This result, [9], shows that ψ can  be made C1,1(R3N) upon multiplication by a factor, universal in the sense that the factor  depends only on N and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We will require an elliptic regularity result, stated below, which will be used in the  proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Beforehand, we clarify the precise form of definitions which we will be using.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Let Ω be open, θ ∈ (0, 1] and k = N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We formally define the θ-H¨older seminorms for a  function f by  [f]θ,Ω = sup  x,y∈Ω  x̸=y  |f(x) − f(y)|  |x − y|θ  ,  [∇kf]θ,Ω = sup  |α|=k  [∂αf]θ,Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ONE-PARTICLE DENSITY MATRIX  7  The space Ck,θ(Ω) is defined as all f ∈ Ck(Ω) where [∇kf]θ,Ω′ is finite for each Ω′  compactly contained in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In addition, the space Ck,θ(Ω) is defined as all f ∈ Ck(Ω)  where [∇kf]θ,Ω is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This space has a norm given by  ∥f∥Ck,θ(Ω) = ∥f∥Ck(Ω) + [∇kf]θ,Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  For open Ω ⊂ Rn we can consider the following elliptic equation,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='34)  Lu := −∆u + c · ∇u + du = g  for some c : Ω → Cn and d, g : Ω → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The corresponding bilinear form for operator L  is defined formally as  L(u, χ) =  �  Ω  �  ∇u · ∇χ + (c · ∇u)χ + duχ  �  dx  for all u ∈ H1  loc(Ω) and χ ∈ C∞  c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We say that a function u ∈ H1  loc(Ω) is a weak  solution to the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='34) in Ω if L(u, χ) =  �  Ω gχ dx for every χ ∈ C∞  c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The following theorem is a restatement of [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1] ([8, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2] is  similar), with additional H¨older regularity which follows from the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let x0 ∈ Rn, R > 0 and c, d, g ∈ L∞(B(x0, R)) and u ∈ H1(B(x0, R))  be a weak solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='34) then for each θ ∈ [0, 1) we have u ∈ C1,θ(B(x0, R)) ∩  H2  loc(B(x0, R)), and for any r ∈ (0, R) we have  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='35)  ∥u∥C1,θ(B(x0,r)) ≤ C(∥u∥L2(B(x0,R)) + ∥g∥L∞(B(x0,R)))  for C = C(n, K, r, R, θ) where  ∥c∥L∞(B(x0,R)) + ∥d∥L∞(B(x0,R)) ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3  Our strategy for the proof will be to choose a suitable function F = F(x), dependent  only on N and Z, such that the function e−Fψ has greater regularity than ψ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Such  a multiplicative factor is frequently called a Jastrow factor in mathematical literature,  and this strategy has been used successfully in, for example, [10], [11] to elucidate reg-  ularity properties of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The function e−Fψ will solve an elliptic equation with bounded  coefficients which behave suitably well under the action of cluster derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Elliptic  regularity will then produce bounds to the cluster derivatives of e−Fψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Such bounds can  then be used to obtain bounds to the cluster derivatives of ψ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Jastrow factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We begin by defining the function  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1)  F(x) = Fc(x) − Fs(x),  8  PETER HEARNSHAW  for x ∈ R3N, where  Fc(x) = −Z  2  �  1≤j≤N  |xj| + 1  4  �  1≤l<k≤N  |xl − xk|,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2)  Fs(x) = −Z  2  �  1≤j≤N  �  |xj|2 + 1 + 1  4  �  1≤l<k≤N  �  |xl − xk|2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3)  The function F was used in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We now detail some basic facts regarding these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Firstly,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4)  ∆Fc = V  where V is the Coulomb potential, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The function Fs has the same behaviour as Fc  at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' However, Fs ∈ C∞(R3N) and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5)  ∂αFs ∈ L∞(R3N)  for all α ∈ N3N  0 , |α| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Finally, the function F obeys  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6)  F, ∇F ∈ L∞(R3N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The function F is used to define the following object, which will be used throughout  the proof:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7)  φ = e−Fψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3) the following elliptic equation can be shown to hold for φ a weak solution,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8)  − ∆φ − 2∇F · ∇φ + (∆F (2) − |∇F|2 − E)φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Since all coefficients are bounded in R3N we can see from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5 that φ ∈ C1,θ(R3N)  for each θ ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Furthermore, by the same theorem, for any pair 0 < r < R we can  find constants C, C′, dependent only on N, Z, E, r, R and θ, such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9)  ∥φ∥C1,θ(B(x,r)) ≤ C ∥φ∥L2(B(x,R)) ≤ C′ ∥φ∥L∞(B(x,R))  for all x ∈ R3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  From the definition of φ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7), along with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6) there exists new constants C < C′,  dependent only on Z and N (hence independent of R > 0), such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10)  C ∥φ∥L∞(B(x,R)) ≤ ∥ψ∥L∞(B(x,R)) ≤ C′ ∥φ∥L∞(B(x,R))  for all x ∈ R3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Derivatives of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Informally, our objective is to take cluster derivatives of the  elliptic equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8) and apply elliptic regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To do so, we require bounds to the  cluster derivatives of the coefficients present in this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This is the objective of the  current section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To begin, we state and prove the following preparatory lemma involving  the distances introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='21), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='25) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ONE-PARTICLE DENSITY MATRIX  9  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For any σ = (σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , σM) ∈ N3M  0  we have for k = 0, 1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='11)  µσ(y)k−|σ| ≤ λσ(y)kλP1(y)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(y)−|σM|  for all y ∈ R3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Furthermore, let β(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , β(n) be an arbitrary collection of multiindices  in N3M  0  such that β(1) + · · · + β(n) = σ then  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='12)  n  �  j=1  λβ(j)(y) ≤ λσ(y)  for all y ∈ R3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The results are trivial in the case of σ = 0 therefore we assume in the following  that σ is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' First, observe for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , M,  µσ(y)−|σj| ≤ λPj(y)−|σj|  µσ(y)1−|σj| ≤ λPj(y)1−|σj|  if σj ̸= 0  by the definition of µσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Now perform the following trivial expansion of the product,  µσ(y)−|σ| = µσ(y)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' µσ(y)−|σM|  ≤ λP1(y)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(y)−|σM|  which proves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='11) for k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For k = 1, consider that for each y we can find l =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , M such that λσ(y) = λPl(y) and σl ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then,  µσ(y)1−|σ| = µσ(y)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' µσ(y)1−|σl| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' µσ(y)−|σM|  ≤ λP1(y)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPl(y)1−|σl| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(y)−|σM|  = λσ(y)λP1(y)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(y)−|σM|  as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Finally we prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' As above, take arbitrary y and a corresponding l such that  λσ(y) = λPl(y) with σl ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For each 1 ≤ j ≤ n we denote the N3  0-components of β(j) as  β(j) = (β(j)  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , β(j)  M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We know β(1)+· · ·+β(n) = σ so in particular, β(1)  l  +· · ·+β(n)  l  = σl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Since σl ̸= 0 there exists at least one 1 ≤ r ≤ n such that β(r)  l  ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Hence by the definition  of λβ(r),  λβ(r)(y) = min{λPs(y) : β(r)  s  ̸= 0, s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , M} ≤ λPl(y) = λσ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The remaining factors λβ(j)(y), for j ̸= r, can each be bounded above by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  The following lemma will be useful in proving results about taking cluster derivatives  of F, as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Later, we will apply it using f as the function |x| for x ∈ R3,  or derivatives thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  10  PETER HEARNSHAW  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let f ∈ C∞(R3\\{0}) and k ∈ N0 be such that for each σ ∈ N3  0 there exists  C such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='13)  |∂σf(x)| ≤ C|x|k−|σ| for all x ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Then for any α ̸= 0 with |α| ≥ k there exists some new C such that for any l, m =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N, the weak derivatives Dα  P(f(xl)) and Dα  P(f(xl −xm)) are both smooth in Σc  α and  obey  |Dα  P(f(xl))|, |Dα  P(f(xl − xm))| ≤ Cqα(x)k−|α|  for all x ∈ Σc  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Take any j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , M with αj ̸= 0, then we have D  αj  Pj(f(xl)) ≡ 0 for each l ∈ P c  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Therefore, for Dα  P(f(xl)) to not be identically zero we require that l ∈ Pj for each j with  αj ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For such l we have xl ̸= 0 since x ∈ Σc  α, and  |xl| ≥ dPj(x)  for each j with αj ̸= 0 by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore, for constant C in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='13), we have  |Dα  P(f(xl))| = |∂α1+···+αMf(xl)| ≤ C|xl|k−|α| ≤ Cqα(x)k−|α|  because |α| ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Similarly, for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , M, with αj ̸= 0 we have D  αj  Pj(f(xl−xm)) ≡  0 if either l, m ∈ Pj or l, m ∈ P c  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore, for Dα  P(f(xl − xm)) to not be identically  zero we require that  (l, m) ∈  �  j : αj̸=0  �  (Pj × P c  j ) ∪ (P c  j × Pj)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  For such (l, m) we have xl ̸= xm since x ∈ Σc  α and  |xl − xm| ≥  √  2 dPj(x)  for each j with αj ̸= 0 by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore, for some constant C′,  |Dα  P(f(xl − xm))| = |∂α1+···+αMf(xl − xm)| ≤ C|xl − xm|k−|α| ≤ C′qα(x)k−|α|  because |α| ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  The following lemma provides pointwise bounds to cluster derivatives of functions  involving F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For any cluster set P and any |σ| ≥ 1 there exists C, which depends on  σ, such that for k = 0, 1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14)  ��Dσ  P∇kF(y)  ��,  ��Dσ  P∇k(eF)(y)  �� ≤ Cλσ(y)1−kλP1(y)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(y)−|σM|  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='15)  ��Dσ  P|∇F(y)|2�� ≤ CλP1(y)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(y)−|σM|  for all y ∈ Σc  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The bound to the first object in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14) also holds when F is replaced by  Fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ONE-PARTICLE DENSITY MATRIX  11  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let τ be the function defined as τ(x) = |x| for x ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then, by definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2)  we can write  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16)  Fc(y) = −Z  2  �  1≤j≤N  τ(yj) + 1  4  �  1≤l<m≤N  τ(yl − ym).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  For each r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N, denote by ∇yr the three-dimensional gradient in the variable yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We then have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='17)  ∇yrFc(y) = −Z  2 ∇τ(yr) + 1  4  N  �  m=1  m̸=r  ∇τ(yr − ym).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We apply the Dσ  P-derivative to each of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='17), using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2 with f = τ  and f = ∇τ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This shows that for k = 0, 1 the functions Dσ  P∇kFc exist and  are smooth on the set Σc  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Furthermore, there exists C, depending on σ, such that  |Dσ  P∇kFc(y)| ≤ Cqσ(y)1−k−|σ| ≤ Cµσ(y)1−k−|σ|  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='18)  for all y ∈ Σc  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In the second inequality we used that µσ(y) ≤ qσ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Recall that ∇Fs is  smooth with all derivatives bounded, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Since F = Fc − Fs and µσ ≤ 1, we have  that for each σ ̸= 0 and k = 0, 1, there exists C, depending on σ, such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='19)  |Dσ  P∇kF(y)| ≤ Cµσ(y)1−k−|σ|  for all y ∈ Σc  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By a straightforward application of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1, this proves the first  bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14), for k = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We proceed to prove the second bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  For every η ≤ σ we have that  Ση ⊂ Σσ and therefore Dη  P∇kF exists and is smooth in Σc  σ, for k = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We use the  chain rule for weak derivatives to show that Dσ  P(eF) exists in Σc  σ and is equal to a sum  of terms, each of the form  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20)  eF �  1≤j≤n  Dβ(j)  P  F  for some 1 ≤ n ≤ |σ| and some collection 0 ̸= β(j) ∈ N3M  0  for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , n, where  β(1) + · · · + β(n) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , n, we write β(j) = (β(j)  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , β(j)  M ) to denote  the N3  0-components of β(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Similarly, the weak derivative Dσ  P∇(eF) exists in Σc  σ and the  gradient of the general term (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20) is equal to  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='21)  eF ∇F  �  1≤j≤n  Dβ(j)  P  F + eF  n  �  r=1  �  Dβ(r)  P  ∇F  �  1≤j≤n  j̸=r  Dβ(j)  P  F  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We will now find bounds for the above expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To do this we will use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='19) to  bound cluster derivatives of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This, along with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='12) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1, allows  12  PETER HEARNSHAW  us to bound the following product  ���  n  �  j=1  Dβ(j)  P  F(y)  ��� ≤ C  n  �  j=1  µβ(j)(y)1−|β(j)|  ≤ C  n  �  j=1  λβ(j)(y)λP1(y)−|β(j)  1 | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(y)−|β(j)  M |  = C  �  n  �  j=1  λβ(j)(y)  �� M  �  l=1  λPl(y)−|β(1)  l  | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPl(y)−|β(n)  l  |�  ≤ Cλσ(y)λP1(y)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(y)−|σM|  where C is some constant, depending on σ, and all y ∈ Σc  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We used that for each  l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , M, we have |β(1)  l |+· · ·+|β(n)  l  | = |σl| as a consequence of β(1) +· · ·+β(n) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We now bound the following product in a similar manner using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='19) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  For any r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , n,  ���Dβ(r)  P  ∇F(y)  �  1≤j≤n  j̸=r  Dβ(j)  P  F(y)  ��� ≤ CλP1(y)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(y)−|σM|  for some constant C, depending on σ, and all y ∈ Σc  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Using the fact that eF and ∇F  are bounded in R3N, it is now straightforward to bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='21) appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  This completes the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Finally, we prove the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' It can be shown that the following Leibniz rule  holds  Dσ  P|∇F|2 =  �  β≤σ  �σ  β  �  Dβ  P∇F · Dσ−β  P  ∇F  in Σc  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For every β ≤ σ we can bound Dβ  P∇F by a constant if β = 0, and by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='19)  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This gives some constant C, depending on σ, such that  ��Dσ  P|∇F(y)|2�� ≤ C  �  β≤σ  µβ(y)−|β|µσ−β(y)−|σ|+|β|  for all y ∈ Σc  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The required bound is obtained after an application of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='11) of Lemma  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The following result extends the bounds given in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3 to give bounds to L∞-  norms in balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' It is useful to note that if x ∈ Σc  σ, for σ ̸= 0, then B(x, νλσ(x)) ⊂ Σc  σ  for all ν ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ONE-PARTICLE DENSITY MATRIX  13  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For any |σ| ≥ 1 and any ν ∈ (0, 1) there exists C, depending on σ and ν,  such that for k = 0, 1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='22)  ��Dσ  P∇kF  ��  L∞(B(x,νλσ(x))) ,  ��Dσ  P∇k(eF)  ��  L∞(B(x,νλσ(x)))  ≤ Cλσ(x)1−kλP1(x)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|σM|  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='23)  ��Dσ  P|∇F|2��  L∞(B(x,νλσ(x))) ≤ CλP1(x)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|σM|  for all x ∈ Σc  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The bound to the first norm in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='22) also holds when F is replaced by  Fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Take some x ∈ Σc  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='22), for each y ∈ B(x, νλσ(x)) we have  |λPj(x) − λPj(y)| ≤ |x − y| ≤ νλσ(x) ≤ νλPj(x)  for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , M with σj ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The final inequality above uses the definition of λσ  in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By rearrangement, we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='24)  (1 − ν)λPj(x) ≤ λPj(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Therefore,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='25)  λP1(y)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(y)−|σM| ≤ (1 − ν)−|σ|λP1(x)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|σM|  for all y ∈ B(x, νλσ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We now prove an analogous inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' First, we see that  λσ(x) = λPl(x) for some 1 ≤ l ≤ M with σl ̸= 0 which will depend on the choice of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We also note that λσ(y) ≤ λPl(y) for all y ∈ R3N which follows from σl ̸= 0 and the  definition of λσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore,  λσ(y)λP1(y)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(y)−|σM| ≤ λPl(y)1−|σl|  M  �  j=1  j̸=l  λPj(y)−|σj|  ≤ (1 − ν)1−|σ|λPl(x)1−|σl|  M  �  j=1  j̸=l  λPj(x)−|σj|  = (1 − ν)1−|σ|λσ(x)λP1(x)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|σM|  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='26)  for all y ∈ B(x, νλσ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In the second step we applied (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The required bounds  then arise from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3 followed by either (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='25) or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='26) as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Cluster derivatives of φ and ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  14  PETER HEARNSHAW  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We introduce the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For cluster set P = (P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , PM) and α ∈ N3M  0  define  φα = Dα  Pφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We have previously obtained estimates for the coefficients, and their derivatives, in equa-  tion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By taking cluster derivatives of this equation, we obtain elliptic equations  whose weak solutions are φα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By a process of rescaling we will apply elliptic regularity to  give quantative bounds to the L∞-norms of φα and ∇φα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' From here it is straightforward  to obtain corresponding bounds for Dα  Pψ and Dα  P∇ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For every |α| ≥ 1 we have φα is a weak solution to  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='27)  − ∆φα − 2∇F · ∇φα + (∆Fs − |∇F|2 − E)φα  =  �  σ≤α  σ̸=0  �α  σ  ��  2Dσ  P∇F · ∇φα−σ − Dσ  P(∆Fs − |∇F|2)φα−σ  �  =: gα  in Σc  α, and therefore φα ∈ C1(Σc  α) ∩ H2  loc(Σc  α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We prove by induction, starting with |α| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let L(·, ·) be the bilinear form  corresponding to the operator acting on φα on the left-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='27), as defined in  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Since φ ∈ H2  loc(R3N) we have φα ∈ H1  loc(R3N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We use integration by parts to  show that, for any χ ∈ C∞  c (Σc  α),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='28)  L(φα, χ) = −L(φ, Dα  Pχ) +  �  Σcα  gαχ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Now, L(φ, Dα  Pχ) = 0 since φ is a weak solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8), hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='27) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By Lemma  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3 and that φ ∈ C1(R3N) we see that gα ∈ L∞  loc(Σc  α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Hence, by Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5, φα ∈  C1(Σc  α) ∩ H2  loc(Σc  α)  Assume the hypothesis holds for all multiindices 1 ≤ |α| ≤ k −1 for some k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Take  some arbitrary multiindex |α| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We will prove the induction hypothesis for α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' First,  we state the useful fact that for any σ ≤ α we have Σσ ⊂ Σα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Take some η ≤ α with  |η| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Notice, then, that φα−η ∈ H2  loc(Σc  α) by the induction hypothesis, and hence  φα ∈ H1  loc(Σc  α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This allows L(φα, χ) to be defined for any χ ∈ C∞  c (Σc  α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Integration  by parts is then used on the Dη  P-derivative, in a similar way to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Along with the  induction hypothesis and further applications of integration by parts, we find that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='27)  holds for φα as a weak solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Regularity for φα is shown in a similar way to the  |α| = 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Indeed, the induction hypothesis can be used to show gα ∈ L∞  loc(Σc  α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We now apply the C1-elliptic regularity estimates of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5, via a scaling proce-  dure, to the equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ONE-PARTICLE DENSITY MATRIX  15  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For all |α| ≥ 1 and 0 < r < R < 1 there exists C, dependent only on  E, Z, N, r and R, such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='29)  ∥∇φα∥L∞(B(x,rλα(x)))  ≤ Cλα(x)−1�  ∥φα∥L∞(B(x,Rλα(x))) + λα(x)2 ∥gα∥L∞(B(x,Rλα(x)))  �  for all x ∈ Σc  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The function gα was given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Fix any x ∈ Σc  α and denote λ = λα(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The proof proceeds via a rescaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Define  w(y) = φα(x + λy)  c(y) = −2∇F(x + λy)  d(y) = ∆Fs(x + λy) − |∇F(x + λy)|2 − E  f(y) = gα(x + λy)  for all y ∈ B(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5 we have that w is a weak solution to  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='30)  − ∆w + λc · ∇w + λ2dw = λ2f  in B(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5), and that λ ≤ 1 by definition, we obtain  λ ∥c∥L∞(B(0,1)) + λ2 ∥d∥L∞(B(0,1)) ≤ K  for some K, dependent only on E, Z and N and in particular is independent of our choice  of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5, with θ = 0, we get some C, dependent only on N, K, r  and R, such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='31)  ∥w∥C1(B(0,r)) ≤ C(∥w∥L∞(B(0,R)) + λ2 ∥f∥L∞(B(0,R))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Finally, we use ∇w(y) = λ∇φα(x + λy) by the chain rule, and rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='31) to give  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  The following proposition uses induction to write estimates for both ∇φα and φα with  a bound involving only the zeroth and first-order derivatives of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2 of the  Appendix is used in the proof to improve the regularity of the second derivative of φ,  and this benefit is passed on through induction to all higher orders of derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The  function f∞( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' φ) was defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For any |α| ≥ 1, any 0 < r < R < 1 and b ∈ [0, 1) there exists C,  dependent on α, r, R and b, such that for k = 0, 1, with k + |α| ≥ 2,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='32)��∇kφα  ��  L∞(B(x,rλα(x))) ≤ Cλα(x)1−kλP1(x)−|α1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|αM|µα(x)bf∞(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' φ)  for all x ∈ Σc  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The inequality also holds for |α| = 1 and k = 0 if we take b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  16  PETER HEARNSHAW  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Suppose |α| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Let l be the index 1 ≤ l ≤ M such that αl ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Then  λPl = λα = µα and Σα = ΣPl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The required bound for k = b = 0 follows from the  definition of cluster derivative, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' When k = 1 and b ∈ [0, 1) the bound follows  directly from Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2 with P = Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  For |α| ≥ 2 we prove by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Assume the hypothesis holds for all multiindices  α with 1 ≤ |α| ≤ m−1 for some m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Take any α with |α| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We prove the k = 0  and k = 1 cases in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' It is useful, here, to state the following fact: for all σ ≤ α we  have Σσ ⊂ Σα and hence λα(x) ≤ λσ(x) for all x ∈ Σc  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The k = 0 case is a straightforward application of the induction hypothesis and is  described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Firstly, for each x ∈ Σc  α it is clear that there exists some 1 ≤ l ≤ M  such that λα(x) = λPl(x) and where αl ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore we can find some η ≤ α with  |η| = 1 and ηj = 0 for each j ̸= l where 1 ≤ j ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Now, from the definition of cluster  derivatives (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16),  ∥φα∥L∞(B(x,rλα(x))) ≤ N ∥∇φα−η∥L∞(B(x,rλα(x))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='33)  We can then use the induction assumption to show existence of C such that  ∥∇φα−η∥L∞(B(x,rλα(x))) ≤ CλPl(x)λP1(x)−|α1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|αM|µα−η(x)bf∞(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' φ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='34)  where we can then use λα(x) = λPl(x) and µα−η(x) ≤ µα(x) to complete the bound  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='32) for our choice of α in the case of k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The k = 1 case follows from the induction hypothesis and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Let r′ =  (r + R)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Firstly, by the definition of gα, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='27), along with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4 we  can find C, depending on α and r′, such that  ∥gα∥L∞(B(x,r′λα(x))) ≤ C  �  σ≤α  σ̸=0  λP1(x)−|σ1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|σM|�  ∥φα−σ∥L∞(B(x,r′λα(x)))  + ∥∇φα−σ∥L∞(B(x,r′λα(x)))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  It follows by the induction hypothesis with b = 0, along with the definition of f∞, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='29),  that we can find some C, depending on α, r′ and R, such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='35)  ∥gα∥L∞(B(x,r′λα(x))) ≤ CλP1(x)−|α1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|αM|f∞(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' φ)  for all x ∈ Σc  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Now, to prove the required bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='32) for k = 1 we apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6  to obtain some constant C, depending on r and r′, such that  ∥∇φα∥L∞(B(x,rλα(x))) ≤ C  �  λα(x)−1 ∥φα∥L∞(B(x,r′λα(x))) + λα(x) ∥gα∥L∞(B(x,r′λα(x)))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We use the k = 0 case, proven above, to bound the L∞-norm of φα in the first term on  the right-hand side of the above inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To the second term, we apply (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='35) and the  simple inequality λα(x) ≤ µα(x)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Together, these give the bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='32) for k = 1 and  our choice of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This completes the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  ONE-PARTICLE DENSITY MATRIX  17  Using the definition φ = e−Fψ, we now obtain bounds to the cluster derivatives of the  eigenfunction ψ using those for the cluster derivatives of φ in the above proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' It is clear that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='31) holds when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore, consider  |α| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We first prove (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='31) for k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Take x ∈ Σc  α, then by the Leibniz rule for  cluster derivatives we have  Dα  Pψ =  �  β≤α  �α  β  �  Dβ  P  �  eF�  φα−β  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='36)  in B(x, rλα(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Now, for each β ≤ α there exists some C, independent of x, such that  ���Dβ  P  �  eF�  φα−β  ���  L∞(B(x,rλα(x))) ≤ Cλα(x)λP1(x)−|α1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|αM|f∞(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  To prove this, we use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7 with b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In addition, if β = 0 we  use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6) and if β = α we use the definition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' If β /∈ {0, α} we use the inequality  λβ(x)λα−β(x) ≤ λα(x), which is obtained from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The required bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='31)  for k = 0 then follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='36) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Take x ∈ Σc  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By the definition F = Fc − Fs, the equality ∇ψ = ψ∇F + eF∇φ and  the Leibniz rule, we have  Dα  P∇ψ = Gα  P + ψDα  P∇Fc  in B(x, rλα(x)), where  Gα  P =  �  β≤α  |β|≥1  �α  β  �  ψβ Dα−β  P  ∇F +  �  β≤α  �α  β  �  ∇φβ Dα−β  P  �  eF�  − ψ Dα  P∇Fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='37)  We bound each term in Gα  P as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Take any β ≤ α with |β| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Notice that  Σβ ⊂ Σα and hence λα(x) ≤ λβ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Now, there exists C, independent of x, such that  ���ψβ Dα−β  P  ∇F  ���  L∞(B(x,rλα(x))) ≤ Cλβ(x)λP1(x)−|α1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|αM|f∞(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='38)  To prove this inequality, we bound derivatives of F using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4 if β ̸= α and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6)  if β = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The derivatives ψβ are then bounded using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='31) with k = 0, which was  proven above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='38) can be bounded as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='33) by using the  simple bound λβ ≤ µα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Next, we can find C, independent of x, such that  ���∇φβ Dα−β  P  �  eF����  L∞(B(x,rλα(x))) ≤ Cµβ(x)bλα−β(x)λP1(x)−|α1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|αM|f∞(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  This is proven using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7 to bound the derivatives ∇φβ, and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4  and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6) to bound derivatives of eF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The right-hand side of the above inequality can be  bounded as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='33) after use of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10) and the inequalities µβ ≤ µα and λα−β ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  In addition, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4 there exists C such that  ��∇φ Dα  P  �  eF���  L∞(B(x,rλα(x))) ≤ Cλα(x)λP1(x)−|α1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|αM|f∞(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  18  PETER HEARNSHAW  Use of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10), as before, and the inequality λα ≤ µα give the correct bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Finally,  the last term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='37) is readily bounded appropriately using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='37),  this completes the proof of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  To prove (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='31) for k = 1 we use (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='32), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='33) and the following inequality,  ∥ψDα  P∇Fc∥L∞(B(x,rλα(x))) ≤ CλP1(x)−|α1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' λPM(x)−|αM|f∞(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R),  which holds for some C as a direct consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Biscaled cutoffs and their associated clusters  In the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1, partial derivatives of γ(x, y) will be written in terms of  integrals involving cluster derivatives of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Cutoff functions are used here to ensure these  cluster derivatives are evaluated only at points in R3N where they exist and where we  can apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The cutoffs introduced in this section are of a similar form to those used in [5], which  themselves are based on those used in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In all these cases, cutoffs enforce bounds on  the distances between pairs of particles on the support of the cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The bounds on  these distances can be scaled using a scaling parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The cutoffs defined here will  involve the points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' the “particles”) x, y and xj, 2 ≤ j ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For a given cutoff, we  will define a corresponding cluster P which is a collection of the particles whose distance  to x is bounded above appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The same is be done for y, with a cluster S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In  using these cutoffs, derivatives of γ(x, y) in the x-variable can naturally be written as  cluster derivatives using the P cluster, and those in the y-variable can be written as  cluster derivatives using the S cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We will also need to take derivatives of γ(x, y) in both x and y simultanously (which  will later be denoted by ∂u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For this, it is natural to consider a larger cluster Q which  contains both P and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The particles in Q are held close to both x and y, but potentially  looser than how particles in P and S are held to x and y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For this reason,  the cutoffs used here will involve two scaling parameters δ and ǫ and hence are called  biscaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Definition of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To begin, take some χ ∈ C∞  c (R), 0 ≤ χ ≤ 1, with  χ(s) =  �  1  if |s| ≤ 1  0  if |s| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1)  for s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For each t > 0 we can define the following two cutoff factors ζt = ζt(z) and  θt = θt(z) by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2)  ζt(z) = χ  �4N|z|  t  �  ,  θt(z) = 1 − ζt(z)  for z ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We have the following support criteria for cutoff factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For any z ∈ R3 and  t > 0,  ONE-PARTICLE DENSITY MATRIX  19  If ζt(z) ̸= 0 then |z| < (2N)−1t,  If θt(z) ̸= 0 then |z| > (4N)−1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Let 0 < δ < (4N)−1ǫ (the use of (4N)−1 is explained in the following lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We  define a biscaled cutoff, which depends on δ and ǫ as parameters, as a function Φ =  Φδ,ǫ(x, y, ˆx) defined by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3)  Φδ,ǫ(x, y, ˆx) =  �  2≤j≤N  g(1)  j (x − xj)  �  2≤j≤N  g(2)  j (y − xj)  �  2≤k<l≤N  fkl(xk − xl)  for x, y ∈ R3 and ˆx ∈ R3N−3, and where g(1)  j , g(2)  j , fkl ∈ {ζδ, θδζǫ, θǫ} for 2 ≤ j ≤ N and  2 ≤ k < l ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In addition, for 2 ≤ k < l ≤ N we define flk = fkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The idea is that on the support of Φ we have upper and/or lower bounds to the  distances between various pairs of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  If we take the particles xj and xk, for  example, we have that ζδ(xj − xk) holds the two particles close relative to a distance  δ, θǫ(xj − xk) holds them apart relative to a distance ǫ, and θδζǫ keeps them within an  intermediate distance apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We use the standard notation 1S to denote the indicator function on a set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' It is  then convenient to define for each t > 0,  1t(z) = 1{(4N)−1t<|z|<(2N)−1t}(z)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4)  1′  t(z) = 1{(4N)−1t<|z|<1}(z)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5)  for z ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' And for each t > 0 we define the function Mt = Mt(x, y, ˆx) by  Mt(x, y, ˆx) =  �  2≤j≤N  1t(x − xj) +  �  2≤j≤N  1t(y − xj) +  �  2≤k<l≤N  1t(xk − xl)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6)  for x, y ∈ R3 and ˆx ∈ R3N−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The cutoffs, Φ, defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3) form a partition of unity as shown in the following  lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' There exists a finite collection {Φ(j)}J  j=1 of biscaled cutoffs (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3), with  integer J depending only on N, such that whenever 0 < δ ≤ (4N)−1ǫ we have  J  �  j=1  Φ(j)(x, y, ˆx) = 1  for all x, y ∈ R3, ˆx ∈ R3N−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' First, we claim that for all z ∈ R3,  θδ(z)θǫ(z) = θǫ(z),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7)  ζδ(z)θǫ(z) = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8)  ζδ(z)ζǫ(z) = ζδ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9)  20  PETER HEARNSHAW  This claim, along with the definitions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2), shows that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10)  1 =  �  ζδ(z) + θδ(z)  ��  ζǫ(z) + θǫ(z)  �  = ζδ(z) + θδ(z)ζǫ(z) + θǫ(z)  for all z ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8) we need only consider z such that θǫ(z) ̸= 0, in  which case |z| > (4N)−1ǫ ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For such z we therefore have θδ(z) = 1, by the definition  of θδ, which proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8), if we also have ζδ(z) ̸= 0, then |z| < (2N)−1δ by the  support criteria for ζδ, giving a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9) we need only consider z such  that ζδ(z) ̸= 0, in which case |z| < (2N)−1δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This gives 4N|z|ǫ−1 < (2N)−1 and hence,  by definition, ζǫ(z) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For any A, B ⊂ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N} with A ∩ B = ∅ we define  τA,B(x) =  �  j∈A  ζδ(x1 − xj)  �  j∈B  (θδζǫ)(x1 − xj)  �  j∈{2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=',N}\\(A∪B)  θǫ(x1 − xj)  and therefore  �  A⊂{2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=',N}  B⊂{2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=',N}\\A  τA,B(x) =  �  2≤j≤N  �  ζδ(x1 − xj) + (θδζǫ)(x1 − xj) + θǫ(x1 − xj)  �  = 1  for all x ∈ R3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let Ξ = {(j, k) : 2 ≤ k < l ≤ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For each subset Y, Z ⊂ Ξ with  Y ∩ Z = ∅ we define  TY,Z(ˆx) =  �  (j,k)∈Y  ζδ(xj − xk)  �  (j,k)∈Z  (θδζǫ)(xj − xk)  �  (j,k) ∈ Ξ\\(Y ∪Z)  θǫ(xj − xk)  and therefore  �  Y ⊂ Ξ  Z ⊂ Ξ\\Y  TY,Z(ˆx) =  �  2≤j<k≤N  �  ζδ(xj − xk) + (θδζǫ)(xj − xk) + θǫ(xj − xk)  �  = 1  for all ˆx ∈ R3N−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Overall,  �  A⊂{2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=',N}  B⊂{2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=',N}\\A  �  C⊂{2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=',N}  D⊂{2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=',N}\\C  �  Y ⊂ Ξ  Z ⊂ Ξ\\Y  τA,B(x, ˆx)τC,D(y, ˆx)TY,Z(ˆx) = 1,  which is a sum of biscaled cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Clusters corresponding to Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We now introduce three clusters corresponding to each Φ, defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To define  these clusters we first introduce index sets based solely on the choice of cutoff factors  g(1)  j , g(2)  j  and fkl in the definition of Φ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The index sets and clusters are therefore  not dependent on x, y, ˆx nor on the scaling parameters δ and ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ONE-PARTICLE DENSITY MATRIX  21  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We define the index set L ⊂ {(j, k) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N}2 : j ̸= k} as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We have (j, k) ∈ L if fjk ̸= θǫ for j, k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Also (1, j), (j, 1) ∈ L if  (g(1)  j , g(2)  j ) ̸= (θǫ, θǫ) for j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Furthermore, we define two more index sets J, K ⊂ {(j, k) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N}2 : j ̸= k} as  follows,  We have (j, k) ∈ J if fjk = ζδ for j, k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Also (1, j), (j, 1) ∈ J if g(1)  j  = ζδ  for j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We have (j, k) ∈ K if fjk = ζδ for j, k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Also (1, j), (j, 1) ∈ K if  g(2)  j  = ζδ for j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  These index sets obey J, K ⊂ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For an arbitrary index set I ⊂ {(j, k) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N}2 :  j ̸= k} we say that two indices j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N} are I-linked if either j = k, or (j, k) ∈ I,  or there exist pairwise distinct indices j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , js for 1 ≤ s ≤ N − 2, all distinct from j  and k such that (j, j1), (j1, j2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , (js, k) ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The cluster Q = Q(Φ) is defined as the set of all indices L-linked to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The cluster  P = P(Φ) is defined as the set of all indices J-linked to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The cluster S = S(Φ) is  defined as the set of all indices K-linked to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Since J, K ⊂ L we see that P, S ⊂ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Support of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Continuing, the clusters P, S and Q will be understood as P(Φ), S(Φ)  and Q(Φ), respectively, when the biscaled cutoff Φ is unambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The following lemma  shows that on the support of Φ, the cluster P ∗ = P\\{1} represents a set of particles xj,  j ∈ P ∗, which are close to x, and the cluster S∗ = S\\{1} represents a set of particles  xj, j ∈ S∗, close to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In both cases this closeness is with respect to the parameter δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The cluster Q∗ = Q\\{1} (recall P, S ⊂ Q) represents a set of particles xj, j ∈ Q∗, close  to both x and y, albeit held potentially looser since this closeness is with respect to the  larger parameter ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let δ ≤ (4N)−1ǫ and δ ≤ |x − y| ≤ 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then Φ(x, y, ˆx) = 0 unless  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='11)  |x − xk| < δ/2 < |y − xk|  for  k ∈ P ∗  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='12)  |y − xk| < δ/2 < |x − xk|  for  k ∈ S∗  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='13)  |x − xk|, |y − xk| < ǫ/2  for  k ∈ Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Therefore, if Φ(x, y, · ) is not identically zero for each x, y ∈ R3 with δ ≤ |x − y| ≤ 2δ,  then P ∗ ∩ S∗ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By definition, if k ∈ P ∗ either g(1)  k  = ζδ or there exist pairwise distinct j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , js ∈  {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N} with 1 ≤ s ≤ N − 2 such that g(1)  j1 = ζδ and fj1,j2, fj2,j3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , fjs,k = ζδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In  the former case, support criteria for ζδ(x − xk) gives |x − xk| < (2N)−1δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In the latter  case, support conditions give |x − xj1|, |xj1 − xj2|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , |xjs − xk| < (2N)−1δ and so by  the triangle inequality |x − xk| < δ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Now, |y − xk| ≥ |x − y| − |x − xk| > δ/2 by the  reverse triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The case of k ∈ S∗ is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  22  PETER HEARNSHAW  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Now let k ∈ Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' First, we consider the case where either g(1)  k  ̸= θǫ or g(2)  k  ̸= θǫ, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Without loss, assume g(1)  k  ̸= θǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then either g(1)  k  = ζδ or g(1)  k  = θδζǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Hence by support  criteria we have the inequalities,  |x − xk| < (2N)−1ǫ,  |y − xk| ≤ |x − y| + |x − xk| ≤ 2δ + (2N)−1ǫ ≤ ǫ/N,  which gives the required inequality since N ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Now, suppose that g(1)  k  = g(2)  k  = θǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Then there exist pairwise distinct j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , js ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N} with 1 ≤ s ≤ N − 2 such that  fj1,j2, fj2,j3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , fjs,k ̸= θǫ and either g(1)  j1 ̸= θǫ or g(2)  j1 ̸= θǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' As before, we see that  |x − xj1|, |y − xj1| ≤ ǫ/N  regardless of which (or both) of g(1)  j1 and g(2)  j1 are not θǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' It’s also clear that by support  criteria, |xj1 − xj2|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , |xjs − xk| ≤ (2N)−1ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore, by the triangle inequality,  |x − xk| ≤ |x − xj1| + |xj1 − xj2| + · · · + |xjs − xk| ≤ ǫ  N + ǫ(N − 2)  2N  = ǫ  2,  and similarly we can show |y − xk| ≤ ǫ/2, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Factorisation of biscaled cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let Φ be given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We can define a  partial product of Φ as a function of the form  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14)  Φ′(x, y, ˆx) =  �  j∈T1  g(1)  j (x − xj)  �  j∈T2  g(2)  j (y − xj)  �  (k,l)∈R1  fkl(xk − xl)  where T1, T2 ⊂ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N}, R1 ⊂ {(k, l) : 2 ≤ k < l ≤ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We now define classes of partial products of Φ which corresponding to a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let  T be an arbitrary cluster with 1 ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Φ(x, y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' T) =  �  j∈T ∗  g(1)  j (x − xj)  �  j∈T ∗  g(2)  j (y − xj)  �  k,l∈T ∗  k<l  fkl(xk − xl),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='15)  Φ(x, y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' T c) =  �  k,l∈T c  k<l  fkl(xk − xl),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16)  Φ(x, y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' T, T c) =  �  j∈T c  g(1)  j (x − xj)  �  j∈T c  g(2)  j (y − xj)  �  k∈T ∗  l∈T c  fkl(xk − xl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='17)  Formulae (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='15) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16) are consistent since the former concerns clusters containing  1 and the latter concerns clusters not containing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  If we consider both x and y to adopt the role of particle 1 then these functions can be  interpreted as Φ( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' T) involving pairs of particles in T, Φ( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' T c) involving pairs in T c,  and Φ( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' T, T c) involving pairs where one lies in T and another lies in T c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ONE-PARTICLE DENSITY MATRIX  23  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Given a biscaled cutoff Φ and any cluster T with 1 ∈ T we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='18)  Φ(x, y, ˆx) = Φ(x, y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' T)Φ(x, y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' T, T c)Φ(x, y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' T c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This identity follows from the definitions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='15)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='17) and the equality  �  2≤k<l≤N  fkl(xk − xl) =  �  k,l∈T ∗  k<l  fkl(xk − xl)  �  k∈T ∗  l∈T c  fkl(xk − xl)  �  k,l∈T c  k<l  fkl(xk − xl),  since for any 2 ≤ k < l ≤ N we have fkl = flk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  Let Q = Q(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then the partial product Φ( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Q, Qc) consists of only θǫ cutoff factors,  as shown in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For Q = Q(Φ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='19)  Φ(x, y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Q, Qc) =  �  j∈Qc  θǫ(x − xj)  �  j∈Qc  θǫ(y − xj)  �  k∈Q∗  l∈Qc  θǫ(xk − xl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By the definition of the cluster Q(Φ) we have g(1)  j  = g(2)  j  = θǫ for each j ∈ Qc,  and fkl = θǫ for each k ∈ Q∗ and l ∈ Qc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Derivatives of biscaled cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The definition of cluster derivatives, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16), is  modestly extended to allow action on biscaled cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For any cluster T we define the  following three cluster derivatives which can act on functions of x, y and ˆx, such as Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We set  Dα  x,T = ∂α  x +  �  j∈T ∗  ∂α  xj  and  Dα  y,T = ∂α  y +  �  j∈T ∗  ∂α  xj  for α ∈ N3  0, |α| = 1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20)  Dα  x,y,T = ∂α  x + ∂α  y +  �  j∈T ∗  ∂α  xj  for α ∈ N3  0, |α| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='21)  which are extended to higher order multiindices α ∈ N3  0 by successive application of  first-order derivatives, as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The following lemma gives partial derivative estimates of the cutoff factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We will  require the following elementary result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For each σ ∈ N3  0 and s ∈ R there exists C > 0  such that for any z0 ∈ R3 we get ∂σ  z |z + z0|s ≤ C|z + z0|s−|σ| for all z ∈ R3, z ̸= −z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Recall the function 1t was defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For any σ ∈ N3  0 with |σ| ≥ 1 and any t > 0 there exists C, depending on  σ but independent of t, such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='22)  |∂σζt(z)|, |∂σθt(z)| ≤ Ct−|σ|1t(z)  for all z ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  24  PETER HEARNSHAW  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Without loss we consider the case of θt, the case of ζt being similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  In the  following, χ(j) refers to the j-th (univariate) derivative of the function χ defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Now, since |σ| ≥ 1 the chain rule shows that ∂σθt(z) can be written as a sum of terms  of the form  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='23)  �4N  t  �m  χ(m)�4N|z|  t  �  ∂σ1  z |z| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ∂σm  z |z|  where 1 ≤ m ≤ |σ|, and σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , σm ∈ N3  0 are non-zero multiindices obeying  σ1 + · · · + σm = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Since m ≥ 1 we have that if χ(m)(s) ̸= 0 then s ∈ (1, 2), and therefore for any term  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='23) to be non-zero we require that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='24)  (4N)−1t < |z| < (2N)−1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  By the remark preceeding the current lemma, there exists C, dependent on σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , σm,  such that  ∂σ1  z |z| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ∂σm  z |z| ≤ C|z|m−|σ| ≤ C(4N)|σ|−mtm−|σ|,  using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore, the terms (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='23) can readily be bounded to give the desired  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  We now give bounds for the cluster derivatives (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='21) acting on cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let Φ be any biscaled cutoff and let Q = Q(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then  Dα  x,y,QΦ( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Q) ≡ 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='25)  for all α ∈ N3  0 with |α| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By the chain rule, each function in the product (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='15) for Φ( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Q) has zero deriv-  ative upon action of Dα  x,y,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  Recall that the notion of partial products was defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14) and the function Mt  for t > 0 was defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let δ ≤ (4N)−1ǫ and Φ = Φδ,ǫ be a biscaled cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let Q = Q(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then  for any multiindex α ∈ N3N+3  0  there exists C, dependent on α but independent of δ and  ǫ, such that for any partial products Φ′ of Φ we have  |∂αΦ′(x, y, ˆx)| ≤  �  Cǫ−|α|  if Φ′ = Φ( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Q, Qc)  C  �  ǫ−|α| + δ−|α|Mδ(x, y, ˆx)  �  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='26)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5 gives bounds for the partial derivatives of the function ζδ, θδ, ζǫ, θǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Considering θδζǫ, we apply the Leibniz rule with σ ∈ N3  0, |σ| ≥ 1, to obtain  ∂σ(θδζǫ)(z) =  �  µ≤σ  �σ  µ  �  ∂µθδ(z)∂σ−µζǫ(z)  = ∂σθδ(z) + ∂σζǫ(z),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='27)  ONE-PARTICLE DENSITY MATRIX  25  since, for each µ ≤ σ with µ ̸= 0 and µ ̸= σ we have  ∂µθδ(z) ∂σ−µζǫ(z) ≡ 0  by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5 and the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Now, to evaluate ∂αΦ′ we apply the Leibniz rule to the product (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14) using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='27)  where appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Differentiated cutoff factors are bounded by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='22) and any remaining  undifferentiated cutoff factors are bounded above by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' If Φ′ = Φ( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Q, Qc), all cutoff  factors are of the form θǫ by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4 and therefore we need only use the bounds in  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='22) with t = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  The derivative Dx,y,Q acting on Φ is special in that it contributes only powers of ǫ (and  not δ) to the bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This is shown in the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let δ ≤ (4N)−1ǫ and Φ = Φδ,ǫ be a biscaled cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let Q = Q(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For  any multiindices α ∈ N3  0 and σ ∈ N3N+3  0  there exists C, independent of ǫ and δ, such  that  |∂σDα  x,y,QΦ(x, y, ˆx)| ≤ Cǫ−|α|�  ǫ−|σ| + δ−|σ|Mδ(x, y, ˆx)  �  for all x, y, ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' First, set Φ′ = Φ( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Q) Φ( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Qc) and Φ′′ = Φ( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Q, Qc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We then have  Dα  x,y,QΦ = Φ′ Dα  x,y,QΦ′′  which follows from Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6 and that Φ( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Qc) is not dependent on variables  involved in the Dα  x,y,Q-derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='21), the derivative Dα  x,y,QΦ′′ can  be written as a sum of partial derivatives of the form ∂αΦ′′ where α ∈ N3N+3  0  obeys  |α| = |α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Now, by the Leibniz rule and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7 there exists some constants C and  C′, independent of δ and ǫ, such that  |∂σ(Φ′ ∂αΦ′′)| ≤  �  τ≤σ  �σ  τ  �  |∂τΦ′| |∂σ−τ+αΦ′′|  ≤ C  �  τ≤σ  (ǫ−|τ| + δ−|τ|Mδ)ǫ−|σ|+|τ|−|α|  ≤ C′ǫ−|α|(ǫ−|σ| + δ−|σ|Mδ),  completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1  The idea of the proof is to turn partial derivatives of an integral, such as the density  matrix γ(x, y) weighted with a suitable cutoff, into cluster derivatives under the inte-  gral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For the density matrix we then estimate the resulting integrals involving cluster  derivatives of ψ using the pointwise bounds of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  26  PETER HEARNSHAW  Although Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1 is stated using partial derivatives of the density matrix γ(x, y) in  the x- and y-variables, it is more appropriate to consider directional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Indeed,  we define new variables  u = (x + y)/2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1)  v = (x − y)/2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2)  and consider the ∂α  u ∂β  v -derivatives of the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The ∂u-derivatives act along  the direction parallel to the diagonal and it is found that they do not affect the non-  smoothness we get at the diagonal, regardless of how many of these derivatives are taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The ∂v-derivatives act in the direction perpendicular to the diagonal and these derivatives  are found to contribute to worsening the non-smoothness at the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Consider derivatives of the form ∂α  x ∂β  y γ(x, y) where |α|+|β| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' As discussed in [5], the  case where |α| = |β| = 1 (the mixed derivatives) is particularly well-behaved in contrast  to the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The reason behind this is that when |α| = |β| = 1, differentiation  under the integral leads to both ψ factors being differentiated exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The greater  regularity in this case follows from the well known fact that ψ, ∇ψ ∈ L∞  loc(R3N), first  proven in [12], whereas higher order derivatives of ψ do not have this locally boundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  This is used in the proof in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='34) below, it will be shown that  derivatives of the form ∂σ  v for |σ| = 2 can be written in terms of ∂σ  u and the mixed  derivatives ∂α  x ∂β  y γ(x, y) for some |α| = |β| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The benefit of this identity is that two  v-derivatives (which act to worsen the singularity at the diagonal) have been transformed  into two u-derivatives (which do not worsen the singularity) along with mixed derivatives  which have good regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  This method only works for two v-derivatives, and the  strategy of using cluster derivatives, as described above, must be used in conjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The difficulties encountered in the fifth derivative of the density matrix are described in  a later section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Density matrix notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The proof will require auxiliary functions related to  the density matrix which we introduce now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For l, m ∈ N3  0 with |l|, |m| ≤ 1 define,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3)  γl,m(x, y) =  �  R3N−3 ∂l  xψ(x, ˆx)∂m  y ψ(y, ˆx) dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  In this notation, it is clear that γ = γ0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For any biscaled cutoff Φ, defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3), we  set  γl,m(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Φ) =  �  R3N−3 ∂l  xψ(x, ˆx)∂m  y ψ(y, ˆx)Φ(x, y, ˆx) dˆx,  and define γ( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Φ) = γ0,0( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We will consider the above functions in the variables  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' It is then natural to define for all u, v ∈ R3,  ˜γl,m(u, v) = γl,m(u + v, u − v),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4)  ˜γl,m(u, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Φ) = γl,m(u + v, u − v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5)  ONE-PARTICLE DENSITY MATRIX  27  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Integrals involving f∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The following proposition is a restatement of [5, Lemma  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1] and is proven in that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Notice that the function Mt was defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6) and  has a slightly different form to the corresponding function used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Given R > 0, there exists C such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6)  �  R3N−3 f∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R) dˆx ≤ C ∥ρ∥1/2  L1(B(x,2R)) ∥ρ∥1/2  L1(B(y,2R))  for all x, y ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In addition, given G ∈ L1(R3) there exists C, independent of G, such  that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7)  �  R3N−3  �  |G(xj − xk)| + |G(z − xk)| + |G(xj)|  �  f∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R) dˆx  ≤ C ∥G∥L1(R3) ∥ρ∥1/2  L1(B(x,2R)) ∥ρ∥1/2  L1(B(y,2R))  for all x, y, z ∈ R3, and j, k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N, j ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In particular, for any t > 0 there exists  C, independent of t, such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8)  �  R3N−3 Mt(x, y, ˆx)f∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R) dˆx ≤ Ct3 ∥ρ∥1/2  L1(B(x,2R)) ∥ρ∥1/2  L1(B(y,2R))  for all x, y ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We introduce the following quantities based on the bounded distances (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For any  x, y ∈ R3 and ˆx ∈ R3N−3 define  λ(x, y, ˆx) = min{λP(x, ˆx), λS∗(x, ˆx), λP ∗(y, ˆx), λS(y, ˆx)},  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9)  π(x, y, ˆx) = min{λQ(x, ˆx), λQ(y, ˆx)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10)  Later, we will see that these quantities appear to negative powers when we apply Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3 to clusters derivatives of ψ involving the clusters P, S and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The next lemma will  give conditions for when these quantities can be bounded away from zero on the support  of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Beforehand, we consider an alternative formulae for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Recall that  1 ∈ P, S, Q by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20) we find that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='11)  π(x, y, ˆx) = min{1, |x|, |y|, |xj| : j ∈ Q∗, 2−1/2|x − xk| : k ∈ Qc,  2−1/2|y − xk| : k ∈ Qc, 2−1/2|xj − xk| : j ∈ Q∗, k ∈ Qc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  In the case of P ∗ ∩ S∗ = ∅ we similarly find that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='12)  λ(x, y, ˆx) = min{1, |x|, |y|, |xj| : j ∈ P ∗ ∪ S∗, 2−1/2|x − xk| : k ∈ P c,  2−1/2|y − xk| : k ∈ Sc, 2−1/2|xj − xk| : (j, k) ∈ (P ∗ × P c) ∪ (S∗ × Sc)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  28  PETER HEARNSHAW  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let δ ≤ (4N)−1ǫ, and ǫ ≤ 1 and let Φ = Φδ,ǫ be an arbitrary biscaled  cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let x, y ∈ R3 be such that δ ≤ |x − y| ≤ 2δ, and |x|, |y| ≥ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then there exists a  constant C, dependent only on N, such that  π(x, y, ˆx) ≥ Cǫ  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='13)  λ(x, y, ˆx) ≥ Cδ  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14)  whenever Φ(x, y, ˆx) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In addition, for all b ≥ 0 with b ̸= 3 and R > 0 there exists C,  depending on b and R but independent of δ, ǫ, x and y such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='15)  �  supp Φ(x,y, · )  λ(x, y, ˆx)−bf∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R) dˆx  ≤ C(ǫ−b + hb(δ)) ∥ρ∥1/2  L1(B(x,2R)) ∥ρ∥1/2  L1(B(y,2R))  where, for all t > 0, we define  hb(t) =  �  0  if b < 3  t3−b  if b > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The following corollary will be useful later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' There exists C, depending on R but independent of δ and ǫ, such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16)  2  �  r=1  �  R3N−3 λ(x, y, ˆx)−2f∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)|∇rΦ(x, y, ˆx)| dˆx  ≤ Cǫ−3 ∥ρ∥1/2  L1(B(x,2R)) ∥ρ∥1/2  L1(B(y,2R))  for all x, y ∈ R3 with δ ≤ |x − y| ≤ 2δ and |x|, |y| ≥ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' When r = 1 the bound for the integral is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Using  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14), the integral in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16) for r = 2 can be bounded by some constant  multiplying  �  supp Φ(x,y, · )  �  ǫ−1λ(x, y, ˆx)−2 + δ−3Mδ(x, y, ˆx)  �  f∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R) dˆx  The required bound follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='15) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1 and that  ǫ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2 we need only consider Φ with P ∗ ∩ S∗ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By the  definition of the cluster Q, if j ∈ Q∗ and k ∈ Qc then fjk = θǫ in the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3)  defining Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Similarly, if k ∈ Qc then g(1)  k  = g(2)  k  = θǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Using the support criteria of θǫ we  therefore get  |x − xk|, |y − xk|, |xj − xk| ≥ (4N)−1ǫ  j ∈ Q∗, k ∈ Qc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='17)  ONE-PARTICLE DENSITY MATRIX  29  In addition, if j ∈ Q∗ we get  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='18)  |xj| �� |x| − |x − xj| ≥ ǫ/2  j ∈ Q∗  since for such j we have |x − xj| ≤ ǫ/2 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The lower bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='13) then  follows from the formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='11) for π(x, y, ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In a similar way we consider the clusters P and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Indeed, let j ∈ P ∗, k ∈ P c or j ∈ S∗,  k ∈ Sc, then fjk ̸= ζδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Similarly, if k ∈ P c then g(1)  k  ̸= ζδ, and if k ∈ Sc then g(2)  k  ̸= ζδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Notice that if a cutoff factor is not ζδ then it must either be θδζǫ or θǫ, both of which  are only supported away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore by the support criteria of these factors we  obtain the following inequalities,  |x − xk|, |y − xl| ≥ (4N)−1δ  k ∈ P c, l ∈ Sc  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='19)  |xj − xk| ≥ (4N)−1δ  j ∈ P ∗, k ∈ P c or j ∈ S∗, k ∈ Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20)  We also have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='18) for j ∈ P ∗ ∪S∗ since P, S ⊂ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The lower bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14) then follows  from the formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='12) for λ(x, y, ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We recall the function 1′  δ was defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By the formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='12) we can use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='18)  to obtain some C, depending on b, such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='21)  λ(x, y, ˆx)−b ≤ C  �  ǫ−b +  �  k∈P c  1′  δ(x − xk) |x − xk|−b +  �  k∈Sc  1′  δ(y − xk) |y − xk|−b  +  �  (j,k)∈(P ∗×P c)  ∪(S∗×Sc)  1′  δ(xj − xk) |xj − xk|−b�  ,  where the upper bounds in the indicator functions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5) can be included because 1 lies in  the minimum (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='12) and the lower bounds follow from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='18)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='15)  then follows from the above inequality along with both (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7) of Proposition  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1, where we choose G to be the function G(z) = 1′  δ(z)|z|−b for z ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Differentiating the density matrix - some required bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We collect cer-  tain results which will be used throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let δ ≤ (4N)−1ǫ and ǫ ≤ 1 and  suppose Φ = Φδ,ǫ is a biscaled cutoff as defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' As usual, we denote Q = Q(Φ),  P = P(Φ) and S = S(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let η ∈ N3  0 and ν = (ν1, ν2) ∈ N6  0 be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Firstly, we  define  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='22)  Φ(η,ν)(x, y, ˆx) = Dη  x,y,QDν1  x,PDν2  y,SΦ(x, y, ˆx)  where in the notation it is implicit the clusters used are Q, P and S corresponding to Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8 and the definition of cluster derivatives (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16) it can be shown that for  each η and ν there exists C, independent of δ and ǫ, such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='23)  |Φ(η,ν)(x, y, ˆx)| ≤ Cǫ−|η|�  ǫ−|ν| + δ−|ν|Mδ(x, y, ˆx)  �  for all x, y ∈ R3 and ˆx ∈ R3N−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  30  PETER HEARNSHAW  Now take any x, y ∈ R3 with δ ≤ |x−y| ≤ 2δ and |x|, |y| ≥ ǫ, and suppose Φ(x, y, ˆx) ̸=  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Then, as a consequence of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2 we have both π(x, y, ˆx) and λ(x, y, ˆx) are  positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore, by the definitions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='21),  (x, ˆx) ∈ Σc  Q ∩ Σc  P ∩ Σc  S∗  and  (y, ˆx) ∈ Σc  Q ∩ Σc  P ∗ ∩ Σc  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='24)  This allows us to apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Indeed, for every η ∈ N3  0, ν = (ν1, ν2) ∈ N6  0 and  all R > 0 there exists C, independent of our choice of x, y and ˆx, such that for k = 0, 1,  ��Dη  QDν  {P,S∗}∇kψ(x, ˆx)  �� ≤ Cπ(x, y, ˆx)−|η|λ(x, y, ˆx)−|ν|f∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='25)  ��Dη  QDν  {P ∗,S}∇kψ(y, ˆx)  �� ≤ Cπ(x, y, ˆx)−|η|λ(x, y, ˆx)−|ν|f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='26)  For convenience, we choose a bound which holds for both values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The functions λ  and π are defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Now, let |ν| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3 with b = 1/2 we can write  Dν  {P,S∗}∇ψ(x, ˆx) = Gν  {P,S∗}(x, ˆx) + ψ(x, ˆx)  �  Dν  {P,S∗}∇Fc(x, ˆx)  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='27)  Dν  {P ∗,S}∇ψ(y, ˆx) = Gν  {P ∗,S}(y, ˆx) + ψ(y, ˆx)  �  Dν  {P ∗,S}∇Fc(y, ˆx)  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='28)  for functions Gν  {P,S∗} and Gν  {P ∗,S} which obey  ��Gν  {P,S∗}(x, ˆx)  �� ≤ Cλ(x, y, ˆx)1/2−|ν|f∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='29)  ��Gν  {P ∗,S}(y, ˆx)  �� ≤ Cλ(x, y, ˆx)1/2−|ν|f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='30)  for some C, dependent on ν but independent of the choice of x, y and ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2), for each ν ∈ N6  0 there exists C, independent of x, y and ˆx,  such that  ��Dν  {P,S∗}∇Fc(x, ˆx)  �� +  ��Dν  {P ∗,S}∇Fc(y, ˆx)  �� ≤ Cλ(x, y, ˆx)−|ν|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='31)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Differentiating the density matrix - first and second derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In the  following, let l, m ∈ N3  0 obey |l| = |m| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In standard notation, by ∂l  x1ψ we mean the  l-partial derivative in the first R3 component of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then by differentiation under the  integral,  ∂l  u˜γ(u, v) =  �  R3N−3 ∂l  x1ψ(u + v, ˆx)ψ(u − v, ˆx) dˆx +  �  R3N−3 ψ(u + v, ˆx)∂lx1ψ(u − v, ˆx) dˆx  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='32)  = ˜γl,0(u, v) + ˜γ0,l(u, v),  and similarly,  ∂l  v˜γ(u, v) = ˜γl,0(u, v) − ˜γ0,l(u, v)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='33)  ONE-PARTICLE DENSITY MATRIX  31  for all u, v ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The above equalities are used to obtain a formula relating second order  u-derivatives to second order v-derivatives of ˜γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Omitting the argument (u, v) we get  �  ∂l+m  u  − ∂l+m  v  �  ˜γ = ∂l  u(˜γm,0 + ˜γ0,m) − ∂l  v(˜γm,0 − ˜γ0,m)  =  �  ∂l  u + ∂l  v  �  ˜γ0,m +  �  ∂l  u − ∂l  v  �  ˜γm,0  = 2(˜γl,m + ˜γm,l)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='34)  where the final equality is obtained by differentiation under the integral, as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Differentiating the density matrix - general derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Partial derivatives  of γ are written as linear combinations of integrals involving cluster derivatives of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  One such integral is bounded in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Since it is more involved than the  other such integrals, the proof is postponed until later in the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let δ ≤ (4N)−1ǫ and ǫ ≤ 1 and let Φ = Φδ,ǫ be an arbitrary biscaled cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Let P = P(Φ) and S = S(Φ) obey P ∗ ∩ S∗ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For any α, β ∈ N6  0 with |α| + |β| = 3,  any l, m ∈ N3  0 with |l| = |m| = 1, and any R > 0 there exists C such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='35)  ���  �  R3N−3  �  Dα  {P,S∗}∂l  xψ(x, ˆx)  ��  Dβ  {P ∗,S}∂m  y ψ(y, ˆx)  �  Φ(x, y, ˆx) dˆx  ���  ≤ Cǫ−3 ∥ρ∥1/2  L1(B(x,R)) ∥ρ∥1/2  L1(B(y,R))  for all δ ≤ |x−y| ≤ 2δ and |x|, |y| ≥ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The constant C depends on R but is independent  of δ, ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  In part two of the following lemma the conditions on η, µ, l and m are not the most  general, but for simplicity we restrict ourselves to these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We use notation  of cluster derivatives of Φ from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let δ ≤ (4N)−1ǫ and ǫ ≤ 1 and Φ = Φδ,ǫ be an arbitrary biscaled cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Let η, µ ∈ N3  0 be arbitrary and l, m ∈ N3  0 be such that |l|, |m| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (i) On the set of u, v such that δ/2 ≤ |v| ≤ δ and |u + v|, |u − v| ≥ ǫ, the derivative  ∂η  u∂µ  v ˜γl,m(u, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Φ) is equal to a linear combination of integrals of the form  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='36)  �  R3N−3  �  Dχ1  Q Dα  {P,S∗}∂l  x1ψ(u + v, ˆx)  ��  Dχ2  Q Dβ  {P ∗,S}∂m  x1ψ(u − v, ˆx)  � Φ(χ3,σ)(u + v, u − v, ˆx) dˆx  where α, β, σ ∈ N6  0 and χ = (χ1, χ2, χ3) ∈ N9  0 obey |χ| = |η| and |α|+|β|+|σ| =  |µ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (ii) Furthermore, suppose either  |µ| ≤ 2, or  |µ| = 3, η = 0 and |l| = |m| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  32  PETER HEARNSHAW  Then for all R > 0 we have some C0 such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='37)  |∂η  u∂µ  v ˜γl,m(u, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Φ)| ≤ C0ǫ−|η|−|µ| ∥ρ∥1/2  L1(B(u+v,R)) ∥ρ∥1/2  L1(B(u−v,R))  for all δ/2 ≤ |v| ≤ δ and |u + v|, |u − v| ≥ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The constant C0 depends on R but  is independent of δ, ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof of i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2, we need only consider Φ such that P ∗ ∩ S∗ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For each  choice of u and v we define a u- and v-dependent change of variables for the integral  ˜γlm(u, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Φ) defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  To start, we define two vectors ˆa = (a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , aN), ˆb =  (b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , bN) ∈ R3N−3 by  ak =  �  u  if k ∈ Q∗  0  if k ∈ Qc  bk =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  v  if k ∈ P ∗  −v  if k ∈ S∗  0  if k ∈ (P ∪ S)c,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='38)  and define ˆωu,v = ˆa + ˆb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We then apply a translational change of variables which allows  us to write  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='39)  ˜γlm(u, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Φ) =  �  R3N−3 ∂l  x1ψ(u+v,ˆz+ ˆωu,v)∂m  x1ψ(u − v,ˆz + ˆωu,v)Φ(u+v, u−v,ˆz+ ˆωu,v) dˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We will then apply differentiation under the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Beforehand, we show how such  derivatives will act on each function within the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For a function f and any  r ∈ N3  0 with |r| = 1 we see that by the chain rule  ∂r  u[f(u ± v,ˆz + ˆωu,v)] = Dr  Qf(u ± v,ˆz + ˆωu,v)  ∂r  v[f(u + v,ˆz + ˆωu,v)] = Dr  Pf(u + v,ˆz + ˆωu,v) − Dr  S∗f(u + v,ˆz + ˆωu,v)  ∂r  v[f(u − v,ˆz + ˆωu,v)] = Dr  P ∗f(u − v,ˆz + ˆωu,v) − Dr  Sf(u − v,ˆz + ˆωu,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Applying repeatedly, we obtain for arbitrary σ, ν ∈ N3  0,  ∂σ  u∂ν  v[f(u + v,ˆz + ˆωu,v)] =  �  τ≤ν  cτ,ν  �  Dσ  QDτ  PDν−τ  S∗ f(u + v,ˆz + ˆωu,v)  �  ∂σ  u∂ν  v [f(u − v,ˆz + ˆωu,v)] =  �  τ≤ν  cτ,ν  �  Dσ  QDτ  P ∗Dν−τ  S  f(u − v,ˆz + ˆωu,v)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  where cτ,ν = (−1)|ν|−|τ|�ν  τ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In a similar manner, for the cutoff we use the definitions  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='22) to write  ∂σ  u∂ν  v[Φ(u + v, u − v,ˆz + ˆωu,v)] =  �  τ≤ν  cτ,νΦ(σ,τ,ν−τ)(u + v, u − v,ˆz + ˆωu,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  By differentiating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='39) under the integral, applying the Leibniz rule and reversing the  change of variables, we find that ∂η  u∂µ  v ˜γlm(u, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Φ) is a linear combination of terms of the  required form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ONE-PARTICLE DENSITY MATRIX  33  Proof of ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By part (i) it suffices to prove the required bound for integrals of the form  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='36) with |χ| = |η| and |α| + |β| + |σ| = |µ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Rewriting such integrals in the variables  x = u + v and y = u − v we get  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='40)  �  R3N−3  �  Dχ1  Q Dα  {P,S∗}∂l  x1ψ(x, ˆx)  ��  Dχ2  Q Dβ  {P ∗,S}∂m  x1ψ(y, ˆx)  �  Φ(χ3,σ)(x, y, ˆx) dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Notice that the variables x and y must obey δ ≤ |x − y| ≤ 2δ and |x|, |y| ≥ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Firstly,  we bound the above integral in the case where |α| + |β| ≤ 2 and |α| + |β| + |σ| ≤ 3 for  any |l|, |m| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='23), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='25), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='26) followed by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='13) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2 we can bound  this integral in absolute value by some constant multiplied by  ǫ−|χ|  �  supp Φ(x,y,·)  �  ǫ−|σ| + δ−|σ|Mδ(x, y, ˆx)  �  λ(x, y, ˆx)−|α|−|β|f∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R) dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Selective use of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2 allows us to bound this quantity by some constant  multiplied by  ǫ−|χ|  �  supp Φ(x,y,·)  �  ǫ−|σ|λ(x, y, ˆx)−|α|−|β| + δ−|α|−|β|−|σ|Mδ(x, y, ˆx)  �  f∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R) dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We can use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='15) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2, using that |α| + |β| ≤ 2 by assumption, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8) of  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1 to bound this quantity by some constant multiplied by  ǫ−|α|−|β|−|σ|−|χ| ∥ρ∥1/2  L1(B(x,2R)) ∥ρ∥1/2  L1(B(y,2R))  where it was used that δ ≤ 1 ≤ ǫ−1 and |α| + |β| + |σ| ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' After a return to u, v-  variables, this proves the bound for integrals (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='36) in the case where |α| + |β| ≤ 2  and |α| + |β| + |σ| ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' It remains to bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='40) in the case where |α| + |β| = 3,  |σ| = |χ| = 0 and |l| = |m| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This follows directly from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Take any η, µ, l, m ∈ N3  0 as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then for all R > 0 we have  C such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='41) |∂η  u∂µ  v ˜γl,m(u, v)| ≤ C min{1, |u + v|, |u − v|}−|η|−|µ| ∥ρ∥1/2  L1(B(u+v,R)) ∥ρ∥1/2  L1(B(u−v,R))  for all u, v ∈ R3 obeying 0 < |v| ≤ (4N)−1 min{1, |u + v|, |u − v|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Firstly, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1 there exists a finite collection of biscaled cutoffs, Φ(j),  j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , J, such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='42)  ˜γl,m =  J  �  j=1  ˜γl,m  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Φ(j)  δ,ǫ  �  holds for all choices of 0 < δ ≤ (4N)−1ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  34  PETER HEARNSHAW  Let C0 be the constant from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5 such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='40) holds for each Φ(j), j =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Fix any v0 ̸= 0 and u0 such that |v0| ≤ (4N)−1 min{1, |u0 + v0|, |u0 − v0|} and  set δ0 = |v0| and ǫ0 = min{1, |u0 + v0|, |u0 − v0|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='42),  |∂η  u∂µ  v ˜γl,m(u0, v0)| ≤  J  �  j=1  ��∂η  u∂µ  v ˜γl,m  �  u0, v0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Φ(j)  δ0,ǫ0  ���  ≤ JC0 min{1, |u0 + v0|, |u0 − v0|}−|η|−|µ| ∥ρ∥1/2  L1(B(u0+v0,R)) ∥ρ∥1/2  L1(B(u0−v0,R)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Since C0 does not depend on the choice of δ0 and ǫ0, the constant JC0 does not depend  on the choice of u0 and v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For all α, β ∈ N3  0 with |α| + |β| = 5 and all R > 0 there exists C,  depending on R, such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='43)  |∂α  u∂β  v ˜γ(u, v)| ≤ C min{1, |u + v|, |u − v|}−4 ∥ρ∥1/2  L1(B(u+v,R)) ∥ρ∥1/2  L1(B(u−v,R))  for all u, v ∈ R3 obeying 0 < |v| ≤ (4N)−1 min{1, |u + v|, |u − v|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' First, we consider the case where |β| ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='32) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='33) for when |β| ≤  2 and |β| = 3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The bound then follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6 with |η| + |µ| = 4  and |µ| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Now, consider 4 ≤ |β| ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Take l, m ∈ N3  0, |l| = |m| = 1 be such that  l + m ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='34),  ∂α  u∂β  v ˜γ(u, v) = ∂α+l+m  u  ∂β−l−m  v  ˜γ(u, v) − 2  �  ∂α  u∂β−l−m  v  ˜γl,m(u, v) + ∂α  u∂β−l−m  v  ˜γm,l(u, v)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The first term on the right-hand side has |β| − |l| − |m| ≤ 3 hence the required bound  follows from the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The remaining terms can be bounded using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6  □  The proof of our main theorem is an immediate consequence of this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The proof follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7 with u = (x + y)/2 and  v = (x − y)/2, along with the definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4, we will examine the cluster derivatives  of ψ present in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='35) more closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' In particular, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3 allows us to write such  derivatives in terms of derivatives of Fc and a function, Gα  P, of higher regularity near  certain singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Sign cancellation allows uniform boundedness of the integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='35)  as x and y approach each other, and is more easily handled via derivatives of Fc rather  than those of ψ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Indeed, due to the simple formula definining Fc, it is possible  to characterise all its cluster derivatives explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' A series of steps, mostly involving  integration by parts, will complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  To begin, we use definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2) to write  ∇xFc(x, ˆx) = −Z  2 ∇x|x| + 1  4  �  2≤j≤N  ∇x|x − xj|  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='44)  ONE-PARTICLE DENSITY MATRIX  35  with the formula also holding when x is replaced by y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let P and S be arbitrary clusters  with 1 ∈ P, S and P ∗ ∩ S∗ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let α = (α1, α2) ∈ N6  0, β = (β1, β2) ∈ N6  0 and let  l, m ∈ N3  0 obey |l| = |m| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then,  Dα  {P,S∗}∂l  x|x| =  �  ∂α1+l  x  |x|  if α2 = 0  0  if |α2| ≥ 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='45)  Dβ  {P ∗,S}∂m  y |y| =  �  ∂β2+m  y  |y|  if β1 = 0  0  if |β1| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='46)  In the following, the cluster derivatives in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='47) are understood to act with respect to  the ordered variables (x, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xN) and the cluster derivatives in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='48) are understood  to act with respect to the ordered variables (y, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For later convenience, on the  right-hand side of both formulae, all derivatives in the x- or y-variable are rewritten to  act on xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Now assume |α|, |β| ≥ 1, then  Dα  {P,S∗}∂l  x|x − xj| =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  (−1)|α1|+1∂α1+α2+l  xj  |x − xj|  if |α2| ≥ 1 and j ∈ S∗  0  if |α2| ≥ 1 and j ∈ Sc  (−1)|α1|+1∂α1+α2+l  xj  |x − xj|  if α2 = 0 and j ∈ P c  0  if α2 = 0 and j ∈ P ∗  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='47)  Dβ  {P ∗,S}∂m  y |y − xj| =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  (−1)|β2|+1∂β1+β2+m  xj  |y − xj|  if |β1| ≥ 1 and j ∈ P ∗  0  if |β1| ≥ 1 and j ∈ P c  (−1)|β2|+1∂β1+β2+m  xj  |y − xj|  if β1 = 0 and j ∈ Sc  0  if β1 = 0 and j ∈ S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='48)  In particular, since P ∗ ∩ S∗ = ∅, we have Dα  {P,S∗}∂l  x|x − xj| ≡ 0 unless j ∈ P c and  Dβ  {P ∗,S}∂m  y |y − xj| ≡ 0 unless j ∈ Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We will often use the following elementary fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For each z0 ∈ R3 and η ∈ N3  0 there  exists C, dependent on η but independent of z0, such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='49)  ��∂η  z |z0 − z|  �� ≤ C|z0 − z|1−|η|  for all z ̸= z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Therefore, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='12) we have for each |η| ≥ 1 some C and C′ such that  ��∂η  xj|x − xj|  �� +  ��∂η  xk|y − xk|  �� ≤ C  �  |x − xj|1−|η| + |y − xk|1−|η|�  ≤ C′λ(x, y, ˆx)1−|η|  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='50)  for all 2 ≤ j, k ≤ N if |η| = 1, and all j ∈ P c, k ∈ Sc if |η| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Here, λ(x, y, ˆx) is defined  in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9) using the clusters P and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let P = P(Φ) and S = S(Φ) obey P ∗ ∩ S∗ = ∅, and take any R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Consider |α| + |β| = 3 and |l| = |m| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For |α|, |β| ≥ 1, the integral  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='51)  �  R3N−3  �  Dα  {P,S∗}∂l  xFc(x, ˆx)  �  ψ(x, ˆx)  �  Dβ  {P ∗,S}∂m  y Fc(y, ˆx)  �  ψ(y, ˆx)Φ(x, y, ˆx) dˆx,  36  PETER HEARNSHAW  can be bounded as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For |α| = 3, the integrals  �  R3N−3  �  Dα  {P,S∗}∂l  xFc(x, ˆx)  �  ψ(x, ˆx)∂m  y F(y, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='52)  �  R3N−3  �  Dα  {P,S∗}∂l  xFc(x, ˆx)  �  ψ(x, ˆx)eF(y, ˆx)∂m  y φ(y, ˆx)Φ(x, y, ˆx) dˆx,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='53)  and, for |β| = 3, the integrals  �  R3N−3 ∂l  xF(x, ˆx)ψ(x, ˆx)  �  Dβ  {P ∗,S}∂m  y Fc(y, ˆx)  �  ψ(y, ˆx)Φ(x, y, ˆx) dˆx,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='54)  �  R3N−3 eF(x, ˆx)∂l  xφ(x, ˆx)  �  Dβ  {P ∗,S}∂m  y Fc(y, ˆx)  �  ψ(y, ˆx)Φ(x, y, ˆx) dˆx  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='55)  can each be bounded as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We first prove the bound for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Use of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='44) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='45)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='48) to expand  the integral will produce a linear combination of the following terms, where j ∈ P c and  k ∈ Sc,  ∂α1+l  x  |x|∂β2+m  y  |y|  �  R3N−3 ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx  if α2 = β1 = 0,  ∂α1+l  x  |x|  �  R3N−3 ∂β1+β2+m  xk  |y − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx  if α2 = 0,  ∂β2+m  y  |y|  �  R3N−3 ∂α1+α2+l  xj  |x − xj|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx  if β1 = 0,  �  R3N−3 ∂α1+α2+l  xj  |x − xj|∂β1+β2+m  xk  |y − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Derivatives of |x| and |y| are bounded using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='49) and |x|, |y| ≥ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Now we bound each  integral in absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The first, second and third integrals are readily bounded by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='50) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Finally, for the fourth integral we use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='59) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Now suppose |α| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  First, we prove the bound for the integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Recall  F = Fc − Fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Using this, along with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='44) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='45)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='48) we can expand the integral  ONE-PARTICLE DENSITY MATRIX  37  as linear combination of the following terms, where j ∈ P c and k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' N},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ∂α1+l  x  |x|∂m  y |y|  �  R3N−3 ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx)ψ(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx)Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx) dˆx  if α2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ∂α1+l  x  |x|  �  R3N−3 ∂m  xk|y − xk|ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx)ψ(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx)Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx) dˆx  if α2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ∂α1+l  x  |x|  �  R3N−3 ∂m  xkFs(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx)ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx)ψ(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx)Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx) dˆx  if α2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ∂m  y |y|  �  R3N−3 ∂α1+α2+l  xj  |x − xj|ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx)ψ(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx)Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx) dˆx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  �  R3N−3 ∂α1+α2+l  xj  |x − xj|∂m  xk|y − xk|ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx)ψ(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx)Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx) dˆx  �  R3N−3 ∂α1+α2+l  xj  |x − xj|∂m  xkFs(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx)ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx)ψ(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx)Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' ˆx) dˆx  Derivatives of |x| and |y| are bounded using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='49) and |x|, |y| ≥ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The first three  integrals above are then readily bounded using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1 and that ∇Fs ∈  L∞(R3N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For the fourth and sixth integral we use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='57) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9 with χ ≡ 1  and χ(x, y, ˆx) = ∂m  xkFs(y, ˆx) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Finally, for the fifth integral we use the same  lemma, specifically (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='54) is similar in the case  where |β| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Next, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='44), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='45) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='47) we can rewrite the integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='53) as a linear  combination of the following two terms, where k ∈ P c,  ∂α1+l  x  |x|  �  R3N−3 ψ(x, ˆx)eF(y, ˆx)∂m  y φ(y, ˆx)Φ(x, y, ˆx) dˆx  if α2 = 0,  �  R3N−3 ∂α1+α2+l  xk  |x − xk|ψ(x, ˆx)eF(y, ˆx)∂m  y φ(y, ˆx)Φ(x, y, ˆx) dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Derivatives of |x| are bounded using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='49) and |x| ≥ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The first integral is then bounded  by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10) and finally (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The second integral is bounded  immediately from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='69) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='55) is similar  in the case where |β| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  38  PETER HEARNSHAW  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To prove the required inequality, first consider the case where  |α|, |β| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We can then use both (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='27) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='28) to write  �  R3N−3  �  Dα  {P,S∗}∂l  xψ(x, ˆx)  ��  Dβ  {P ∗,S}∂m  y ψ(y, ˆx)  �  Φ(x, y, ˆx) dˆx  =  �  R3N−3  �  Gα,l  {P,S∗}(x, ˆx)  ��  Dβ  {P ∗,S}∂m  y ψ(y, ˆx)  �  Φ(x, y, ˆx) dˆx  +  �  R3N−3  �  Dα  {P,S∗}∂l  xFc(x, ˆx)  �  ψ(x, ˆx)  �  Gβ,m  {P ∗,S}(y, ˆx)  �  Φ(x, y, ˆx) dˆx  +  �  R3N−3  �  Dα  {P,S∗}∂l  xFc(x, ˆx)  �  ψ(x, ˆx)  �  Dβ  {P ∗,S}∂m  y Fc(y, ˆx)  �  ψ(y, ˆx)Φ(x, y, ˆx) dˆx  We bound each of these integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We start with the final integral on the right-hand side  which is just (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='51) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Next, by using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='25)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='26) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='29)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='31), the  first and second integrals on the right-hand side can be bounded in absolute value by  some constant multiplied by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='56)  �  R3N−3 λ(x, y, ˆx)−5/2f∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)Φ(x, y, ˆx) dˆx  ≤ Cǫ−5/2 ∥ρ∥1/2  L1(B(x,2R)) ∥ρ∥1/2  L1(B(y,2R))  where the inequality above holds for some C by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='15) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This proves the  required bound when |α|, |β| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Now we consider the case where |α| = 3, and hence β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We use that ∇ψ =  ψ∇F + eF∇φ and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='27) to give  �  R3N−3  �  Dα  {P,S∗}∂l  xψ(x, ˆx)  �  ∂m  y ψ(y, ˆx)Φ(x, y, ˆx) dˆx  =  �  R3N−3  �  Gα,l  {P,S∗}(y, ˆx)  �  ∂m  y ψ(y, ˆx)Φ(x, y, ˆx) dˆx  +  �  R3N−3  �  Dα  {P,S∗}∂l  xFc(x, ˆx)  �  ψ(x, ˆx)∂m  y F(y, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx  +  �  R3N−3  �  Dα  {P,S∗}∂l  xFc(x, ˆx)  �  ψ(x, ˆx)eF(y, ˆx)∂m  y φ(y, ˆx)Φ(x, y, ˆx) dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  As before, the first integral on the right-hand side can be bounded in absolute value by  some constant multiplying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The second and third integrals on the right-hand side  are just (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='52) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='53) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The proof of the |β| = 3 case is similar to the  |α| = 3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  We now prove two lemmas which were used to prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let P = P(Φ) and S = S(Φ) obey P ∗ ∩ S∗ = ∅, and take any R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ONE-PARTICLE DENSITY MATRIX  39  (i) Let |η| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let χ = χ(x, y, ˆx) ∈ C∞(R3N+3) such that χ, ∇χ ∈ L∞(R3N+3) (for  example χ ≡ 1 may be chosen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then, for all j ∈ P c, the integral  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='57)  �  R3N−3 ∂η  xj|x − xj|χ(x, y, ˆx)ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx,  and, for all k ∈ Sc, the integral  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='58)  �  R3N−3 ∂η  xk|y − xk|χ(x, y, ˆx)ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx  can be bounded as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (ii) Let |α| + |β| = 3 and |l| = |m| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then for any pair 2 ≤ j, k ≤ N such that  j ∈ P c if |α| ≥ 1 and k ∈ Sc if |β| ≥ 1 we have that the integral  Ij,k =  �  R3N−3 ∂α+l  xj |x − xj|∂β+m  xk  |y − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='59)  can be bounded as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof of i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Take any j ∈ P c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By a product rule for weak derivatives  we know that  ∇xj  �  ψ(x, ˆx)ψ(y, ˆx)  �  =  �  ∇xjψ(x, ˆx)  �  ψ(y, ˆx) + ψ(x, ˆx)  �  ∇xjψ(y, ˆx)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The functions χ and Φ are both smooth so are readily included in such a product rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Take any multiindex τ ≤ η with |τ| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Using integration by parts we get that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='57)  equals  −  �  R3N−3 ∂η−τ  xj |x − xj|∂τ  xj  �  χ(x, y, ˆx)ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx)  �  dˆx  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='60)  which, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='50), can be bounded in absolute value by some constant, depending on χ,  multipling the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16) which is bounded in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='58)  is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof of ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We first prove the case where j ̸= k, where integration by parts is particularly  simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Suppose |α| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Using a strategy similar to the proof of i), we use integration  by parts to obtain  Ij,k = −  �  R3N−3 ∂α  xj|x − xj|∂β+m  xk  |y − xk|∂l  xj  �  ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx)  �  dˆx  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='61)  which can then be bounded using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The case of |β| ≥ 1 is similar except we  apply integration by parts on the ∂m  xk-derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This completes the proof where j ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  For the remainder of the proof we consider the case where k = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Suppose, first, that  k ∈ (S ∪ P)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To simplify calculations we write η = α + l and µ = β + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Notice that  40  PETER HEARNSHAW  |η|, |µ| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore we can find some multiindex µ1 ≤ µ with |µ1| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Integration by  parts then gives  Ik,k = −  �  R3N−3 ∂η+µ1  xk  |x − xk|∂µ−µ1  xk  |y − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx  −  �  R3N−3 ∂η  xk|x − xk|∂µ−µ1  xk  |y − xk|∂µ1  xk  �  ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx)  �  dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='62)  We leave untouched the second integral above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For the first, if |µ|−|µ1| ≥ 1 we can remove  another first-order derivative from |y − xk| by the same procedure - using integration by  parts to give two new terms as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We retain the term where the derivative falls  on ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Whereas on the term where the derivative falls on |x − xk|  we repeat the procedure, so long as there remains a non-trivial derivative on |y − xk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Through this process, we obtain the following formula for Ik,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let T = |µ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then write  µ = �T  i=1 µi for some collection |µi| = 1, where 1 ≤ i ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Furthermore, define  µ<j =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  0  if j = 1  µ1  if j = 2  µ1 + · · · + µj−1  if j ≥ 3  µ>j =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  0  if j = T  µT  if j = T − 1  µj+1 + · · · + µT  if j ≤ T − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Then,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='63)  Ik,k = (−1)T  �  R3N−3 ∂η+µ  xk |x − xk||y − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +  T  �  j=1  (−1)jI(j)  k,k  where  I(j)  k,k =  �  R3N−3 ∂η+µ<j  xk  |x − xk|∂µ>j  xk |y − xk|∂µj  xk  �  ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx)  �  dˆx  Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='50), for each 1 ≤ j ≤ T − 1 we can bound |I(j)  k,k| by some constant multiplying  the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16), which is bounded in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  It remains to bound I(T)  k,k ,  along with the first integral in the formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Starting with the latter, we begin by  expanding |y − xk| = |x − xk| +  �  |y − xk| − |x − xk|  �  and noticing that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='64)  ��|y − xk| − |x − xk|  �� ≤ |x − y| ≤ 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  To bound the first integral in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='63), it then suffices to bound the two integrals:  δ  �  R3N−3  ��∂η+µ  xk |x − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx)  �� dˆx,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='65)  �  R3N−3 ∂η+µ  xk |x − xk||x − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='66)  ONE-PARTICLE DENSITY MATRIX  41  We start with the first of these two integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Since k ∈ (P ∪ S)c we can use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='50) to  show that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='65) is bounded by some constant multiplied by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='67)  δ  �  R3N−3 λ(x, y, ˆx)−4f∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)Φ(x, y, ˆx) dˆx  ≤ Cδ(ǫ−4 + δ−1) ∥ρ∥1/2  L1(B(x,2R)) ∥ρ∥1/2  L1(B(y,2R))  where the bound holds by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We can then use the simplification δ(ǫ−4 +δ−1) ≤  Cǫ−3 for some new constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Before looking at (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='66), we next bound I(T)  k,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='64)  we get  ��I(T)  k,k  �� ≤ 2δ  �  R3N−3  ��∂η+µ<T  xk  |x − xk| ∂µT  xk  �  ψ(x, ˆx)ψ(y, ˆx)  �  Φ(x, y, ˆx)  �� dˆx  + 2δ  �  R3N−3  ��∂η+µ<T  xk  |x − xk| ψ(x, ˆx)ψ(y, ˆx)  �  ∂µT  xk Φ(x, y, ˆx)  ��� dˆx  +  �  R3N−3 |x − xk|  ��∂η+µ<T  xk  |x − xk| ∂µT  xk  �  ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx)  ��� dˆx  We bound each integral on the right-hand side of this inequality in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' All will require  use of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='50) to bound derivatives of |x − xk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The first and third integrals can be  bounded by some constants multiplying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='67) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The second term  is bounded by some constant multiplied by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='68)  δ  �  R3N−3 λ(x, y, ˆx)−3f∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' R)|∇Φ(x, y, ˆx)| dˆx  which, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14), can be bounded by some constant multiplying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Finally, it remains to bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We simplify the calculation by denoting σ = η +µ  and writing σ = σ1 + · · · + σ5 for |σi| = 1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Define  σ<j =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  0  if j = 1  σ1  if j = 2  σ1 + · · · + σj−1  if 3 ≤ j ≤ T  σ>j =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  0  if j = 5  σ5  if j = 4  σj+1 + · · · + σ5  if 1 ≤ j ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We now apply the same method used above, that is, we transfer successive first order  derivatives via integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To begin, we apply integration by parts to transfer  ∂σ1  xk from ∂σ  xk|x − xk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We leave as a remainder the term where the derivative falls on  ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' However, for the term where the derivative falls on |x − xk| we  continue the procedure to now remove ∂σ2  xk from ∂σ−σ1  xk  |x − xk| using integration by parts  again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Since |σ| = 5 is odd, the result after this procedure has occured five times is  that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='66) is equal to minus the same integral plus remainder terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This explains the  42  PETER HEARNSHAW  (1/2)-factor in the following formula,  �  R3N−3 ∂σ  xk|x − xk||x − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx  = 1  2  5  �  j=1  (−1)j  �  R3N−3 ∂σ>j  xk |x − xk|∂σ<j  xk |x − xk|∂σj  xk  �  ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx)  �  dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='49)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='50), each integral on the right-hand side is bounded by some constant  multiplying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='16) which is bounded according to Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This completes the proof  in the case where k ∈ (S ∪ P)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  It remains to bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='59) for when k = j but where k ∈ P ∗ or k ∈ S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' First, suppose  k ∈ P ∗ and hence, by the hypothesis, we need only consider α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This implies that  k ∈ Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To bound the integral, we first apply integration by parts to obtain  Ik,k = −  �  R3N−3 |x − xk| ∂β+l+m  xk  |y − xk| ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx  −  �  R3N−3 |x − xk| ∂β+m  xk  |y − xk| ∂l  xk  �  ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx)  �  dˆx  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2 we have |x−xk| < δ/2 when Φ(x, y, ˆx) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Using this, along with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='50),  we get  |Ik,k| ≤ δ  2  �  R3N−3  ��∂β+l+m  xk  |y − xk| ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx)  �� dˆx  + δ  2  �  R3N−3  ��∂β+m  xk  |y − xk| ∂l  xk  �  ψ(x, ˆx)ψ(y, ˆx)  �  Φ(x, y, ˆx)  �� dˆx  + δ  2  �  R3N−3  ��∂β+m  xk  |y − xk| ψ(x, ˆx)ψ(y, ˆx)  �  ∂l  xkΦ(x, y, ˆx)  ��� dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  On the right-hand side of the above inequality, the first and second terms can be bounded  by some constant multiplying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The third term is bounded by some constant  multiplying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The case of k = j with k ∈ S∗ is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let P = P(Φ) and S = S(Φ) obey P ∗ ∩ S∗ = ∅, and take any R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Let  η, l, m ∈ N3  0 obey |η| = 4 and |l| = |m| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then, for each j ∈ P c, the integral  �  R3N−3 ∂η  xj|x − xj|ψ(x, ˆx)eF(y, ˆx)∂m  y φ(y, ˆx)Φ(x, y, ˆx) dˆx  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='69)  and, for each k ∈ Sc, the integral  �  R3N−3 eF(x, ˆx)∂l  xφ(x, ˆx)∂η  xk|y − xk|ψ(y, ˆx)Φ(x, y, ˆx) dˆx  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='70)  can be bounded as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ONE-PARTICLE DENSITY MATRIX  43  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We prove the bound for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The case of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='70) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' First, take some  function χ ∈ C∞  c (R), 0 ≤ χ ≤ 1, with  χ(t) =  �  1  if |t| ≤ 1  0  if |t| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Furthermore, define χR(t) = χ(t/R) for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  It suffices to bound the following two integrals  �  R3N−3  �  1 − χR(|x − xj|)  �  ∂η  xj|x − xj|ψ(x, ˆx)eF(y, ˆx)∂m  y φ(y, ˆx)Φ(x, y, ˆx) dˆx  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='71)  �  R3N−3 χR(|x − xj|)∂η  xk|x − xj|ψ(x, ˆx)eF(y, ˆx)∂m  y φ(y, ˆx)Φ(x, y, ˆx) dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='72)  First, we see that when χR(|x − xj|) ̸= 1 we have |x − xj| > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' This, along with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6)  and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='49) gives some constant C, depending on R, such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='71) can be bounded in  absolute value by  �  R3N−6  �  {xj:|x−xj|>R}  ��∂η  xj|x − xj|ψ(x, ˆx)eF(y, ˆx)∇φ(y, ˆx)Φ(x, y, ˆx)  �� dxjdˆx1j  ≤ C  �  R3N−3  ��ψ(x, ˆx)∇φ(y, ˆx)  �� dˆx,  which itself can be bounded using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10) by some constant multiplying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6) with,  for example, the same R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The relevant bound then follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1, and  that ǫ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Recall the notation introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='11)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='14), namely we can write  (y, x, ˆx1j) = (y, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xj−1, x, xj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , xN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Using ∂m  y φ(y, ˆx) = ∂m  y φ(y, x, ˆx1j) +  �  ∂m  y φ(y, ˆx) − ∂m  y φ(y, x, ˆx1j)  �  it follows that in order  to bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='72) it suffices to bound the following two integrals,  �  R3N−3 χR(|x − xj|)∂η  xj|x − xj|ψ(x, ˆx)eF(y, ˆx)∂m  y φ(y, x, ˆx1j)Φ(x, y, ˆx) dˆx,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='73)  �  R3N−3 χR(|x − xj|)∂η  xj|x − xj|ψ(x, ˆx)eF(y, ˆx)  �  ∂m  y φ(y, ˆx) − ∂m  y φ(y, x, ˆx1j)  �  Φ(x, y, ˆx) dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='74)  Notice that when χR(|x − xj|) ̸= 0 we have |x − xj| < 2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Take any θ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Then  since φ ∈ C1,θ(R3N), we have local boundedness and local θ-H¨older continuity of ∇φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Therefore, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10) there exists a constant C such that, when |x − xj| < 2R,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='75)  |∂m  y φ(y, x, ˆx1j)| ≤ ∥∇φ∥L∞(B((y,ˆx),2R)) ≤ Cf∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' 4R)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='76)  ��∂m  y φ(y, ˆx) − ∂m  y φ(y, x, ˆx1j)  �� ≤ |x − xj|θ[∇φ]θ,B((y,ˆx),2R) ≤ C|x − xj|θf∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' 4R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  44  PETER HEARNSHAW  The constant C depends on R and θ but is independent of x, y and ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Integration by parts in the variable xj is used in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='73) to remove a single derivative  from ∂η  xj|x−xj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Since ∂m  y φ(y, x, ˆx1j) has no dependence on xj, this process avoids taking  a second derivative of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Take any τ ≤ η with |τ| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='73) can therefore be  rewritten as  −  �  R3N−3 ∂τ  xj  �  χR(|x − xj|)  �  ∂η−τ  xj |x − xj| ψ(x, ˆx) eF(y, ˆx) ∂m  y φ(y, x, ˆx1j) Φ(x, y, ˆx) dˆx  −  �  R3N−3 χR(|x − xj|) ∂η−τ  xj |x − xj| ∂m  y φ(y, x, ˆx1j) ∂τ  xj  �  eF(y, ˆx)ψ(x, ˆx)Φ(x, y, ˆx)  �  dˆx  which, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='50) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='75), can be bounded in absolute value by  C  �  R3N−3 λ(x, y, ˆx)−2f∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' 4R)f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' 4R)  �  Φ(x, y, ˆx) + |∇Φ(x, y, ˆx)|  �  dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  for some C depending on R and our choice of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The relevant bound then follows by  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='49), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='50) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='76), the integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='74) can then be bounded in  absolute value by some constant multiplied by  �  R3N−3 λ(x, y, ˆx)−3+θf∞(x, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' 4R)f∞(y, ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' 4R)Φ(x, y, ˆx) dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  The relevant bound then follows by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2 with b = −3 + θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Second derivatives of φ  Fix some function χ ∈ C∞  c (R), 0 ≤ χ ≤ 1, with  χ(t) =  �  1  if |t| ≤ 1  0  if |t| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We also set  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1)  g(x, y) = (x · y) ln  ���  |x|2 + |y|2�  for x, y ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For each x ∈ R3N we can then define the function  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2)  G(x) = K0  �  1≤j<k≤N  χ(|xj|)χ(|xk|)g(xj, xk)  where K0 = Z(2 − π)(12π)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The function G was first introduced by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Fournais, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Hoffmann-Ostenhof, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Ø.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Sørensen in [9] to improve the regularity of φ via  a multiplicative factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Define  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3)  φ′ := e−Gφ = e−(G+F )ψ = e−(G+Fc−Fs)ψ  Then, in the same paper it was shown that φ′ ∈ C1,1(R3N) and the factor e−(G+F ) is  optimal in the sense that no other multiplicative factor, depending on N and Z but  ONE-PARTICLE DENSITY MATRIX  45  not on ψ or E, can produce greater regularity than C1,1(R3N) for all eigenfunctions ψ  obeying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The quantitative result is the following, which is an adapted version of  [9, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For all 0 < r < R < 1 we have a constant C(r, R), depending on r and  R but independent of ψ, such that  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4)  ∥φ′∥W 2,∞(B(x,r)) ≤ C(r, R) ∥φ′∥L∞(B(x,R)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  for all x ∈ R3N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We use the standard result that if u ∈ C1,1(B) then u ∈ W 2,∞(B) and  ∥Diju∥L∞(B) ≤ [u]1,B, see for example [13]  Our objective will be to use this theorem to obtain pointwise bounds for the second  derivatives of φ, which we do not expect to be bounded since φ lacks the additional factor  e−G present in the definition of φ′, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' We obtain the following as a corollary of the  above theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For all 0 < r < R < 1 and b > 0 there exists C, depending on r, R  and b but independent of ψ, such that for any non-empty cluster P and any η ∈ N3  0 with  |η| = 1,  ∥Dη  P∇φ∥L∞(B(x,rλP (x))) ≤ CλP(x)−b ∥φ∥L∞(B(x,R))  for all x ∈ Σc  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To begin, we obtain bounds for derivatives of g(x, y) and G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' It can be seen  that g ∈ C1,θ(R6) for all θ ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' For the second derivatives, let α, β ∈ N3  0 obey  |α| + |β| = 2, then there exists C such that  |∂α  x ∂β  y g(x, y)| ≤  �  C +  �� ln  �  |x|2 + |y|2���  if |α| = |β| = 1  C  otherwise  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='5)  for all x, y ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' It follows that  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6)  G, ∇G ∈ L∞(R3N),  and given b > 0 there exist constants C and C′, only the latter depending on b, such  that for any η ∈ N3  0 with |η| = 1, and k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , N, we have  |∂η  xk∇G(y)| ≤ C  �  1 +  N  �  l=1  l̸=k  χ(|yk|)χ(|yl|)  �� ln  �  |yk|2 + |yl|2���  �  ≤ C′�  1 + |yk|−b�  for all y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' , yN) ∈ R3N with yk ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Using the above inequality for every k ∈ P  and the definition of cluster derivatives, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='15), we can obtain some C, depending on b,  46  PETER HEARNSHAW  such that  |Dη  P∇G(y)| ≤ CλP(y)−b  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7)  for all y ∈ Σc  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Here, we also used that λP ≤ 1, the definition of λP in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='21), and the  formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Now, take some x ∈ Σc  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' As in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='4, we use (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='22) to show that  (1 − r)λP(x) ≤ λP(y) for each y ∈ B(x, rλP(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Therefore, for C as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='7), we have  ∥Dη  P∇G∥L∞(B(x,rλP (x))) ≤ C(1 − r)−bλP(x)−b  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8)  for all x ∈ Σc  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  We now in a position to consider derivatives of φ = eGφ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Firstly, we have ∇φ =  eGφ′∇G + eG∇φ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' And therefore the following formula holds for each η ∈ N3  0, |η| = 1,  Dη  P∇φ =  �  Dη  P∇G + Dη  PG ∇G  �  φ + eGDη  Pφ′ ∇G + eGDη  PG ∇φ′ + eGDη  P∇φ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  Taking the norm and using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='8) we can then obtain C such that  ∥Dη  P∇φ∥L∞(B(x,rλP (x))) ≤ C  �  λP(x)−b ∥φ∥L∞(B(x,rλP (x))) + ∥φ′∥W 2,∞(B(x,rλP (x)))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9)  for all x ∈ Σc  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' To the second term in the above bound we may then use Theorem  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='1 with constant C(r, R), followed by use of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='6) to obtain another constant C′, also  dependent on r, R but independent of x, such that  ∥φ′∥W 2,∞(B(x,rλP (x))) ≤ C(r, R) ∥φ′∥L∞(B(x,R)) ≤ C′ ∥φ∥L∞(B(x,R)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10)  Together, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='9) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='10) complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  □  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' The author would like to thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Sobolev for helpful dis-  cussions in all matters of the current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
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+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
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+page_content=' The diagonal behaviour of the one-particle coulombic density  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
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+page_content=' Off-diagonal derivative discontinuities in the reduced density matrices of electronic  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Phys, 153(154108), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
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+page_content=' Cioslowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Reverse engineering in quantum chemistry: How to reveal the fifth-order off-diagonal  cusp in the one-electron reduced density matrix without actually calculating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Int J Quantum  Chem, 122(8)(e26651), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='  ONE-PARTICLE DENSITY MATRIX  47  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content=' Fournais and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='Ø.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
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+page_content='  Department of Mathematics, University College London, Gower Street, London,  WC1E 6BT UK  Email address: peter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='hearnshaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='18@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'}
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