diff --git "a/59E1T4oBgHgl3EQfBQIH/content/tmp_files/2301.02848v1.pdf.txt" "b/59E1T4oBgHgl3EQfBQIH/content/tmp_files/2301.02848v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/59E1T4oBgHgl3EQfBQIH/content/tmp_files/2301.02848v1.pdf.txt" @@ -0,0 +1,3481 @@ +arXiv:2301.02848v1 [math-ph] 7 Jan 2023 +BOUNDEDNESS OF THE FIFTH OFF-DIAGONAL DERIVATIVE FOR +THE ONE-PARTICLE COULOMBIC DENSITY MATRIX +PETER HEARNSHAW +Abstract. 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. +1. 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. For simplicity we restrict ourselves +to the case of an atom (N0 = 1), although all results readily generalise to the molecular +case. +The electrons have coordinates x = (x1, . . . , xN), xk ∈ R3, k = 1, . . . , N, and +the nucleus has charge Z > 0 and its position fixed at the origin. The corresponding +Schr¨odinger operator is +(1.1) +H = −∆ + V +where ∆ = �N +k=1 ∆xk is the Laplacian in R3N, i.e. ∆xk refers to the Laplacian applied +to the variable xk, and V is the Coulomb potential given by +(1.2) +V (x) = − +N +� +k=1 +Z +|xk| + +� +1≤j 0 define +hb(t) = +� +tmin{0,5−b} +if b ̸= 5 +log(t−1 + 2) +if b = 5. +We denote N0 = N ∪ {0}. For all R > 0 and α, β ∈ N3 +0 with |α|, |β| ≥ 1 there exists C +such that +(1.7) +|∂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 +(1.8) +|∂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. The notation ∂α +x refers to the α-partial +derivative in the x variable. The constant C depends on α, β, R, N and Z. The right- +hand side is finite because ψ is normalised and hence ρ ∈ L1(R3). The bounded first +derivative at the nucleus reflects local Lipschitz continuity of γ on R6. In addition, there +is local boundedness of up to four derivatives at the diagonal with at worst a logarithmic +singularity for the fifth derivative. The purpose of this current paper is to show local +boundedness of the fifth derivative at the diagonal. +The existence of the fifth-order cusp at the diagonal was previously demonstrated in +[6] (see also [7]). 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) +(1.9) +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. +Our main result is as follows. +Theorem 1.1. Let ψ be an eigenfunction of (1.3). Define m(x, y) = min{1, |x|, |y|}. +Then for all |α| + |β| = 5 and R > 0 we have C, depending on R and also on Z, N and +E, such that +(1.10) +|∂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). +Remark 1.2. +(1) Theorem 1.1 naturally extends to the case of a molecule with +several nuclei whose positions are fixed. The modifications are straightforward. +(2) The bound (1.10) naturally complements (1.7) and (1.8) for |α| + |β| ̸= 5. +(3) As a consequence of [5, Proposition 7.1], the inequality (1.10) shows that γ ∈ +C4,1 +loc +� +(R3\{0}) × (R3\{0}) +� +. +Notation. As mentioned earlier, we use standard notation whereby x = (x1, . . . , xN) ∈ +R3N, xj ∈ R3, j=1, . . . , N, and where N is the number of electrons. In addition, define +for 1 ≤ j, k ≤ N, j ̸= k, +ˆxj = (x1, . . . , xj−1, xj+1, . . . , xN) +(1.11) +ˆxjk = (x1, . . . , xj−1, xj+1, . . . , xk−1, xk+1, . . . , xN) +(1.12) +with obvious modifications if either j, k equals 1 or N, and if k < j. We define ˆx = ˆx1, +which will be used throughout. 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, . . . , xj−1, x, xj+1, . . . , xN), +(1.13) +(x, y, ˆxjk) = (x1, . . . , xj−1, x, xj+1, . . . , xk−1, y, xk+1, . . . , xN). +(1.14) +In this way, x = (xj, ˆxj) = (xj, xk, ˆxjk). + +4 +PETER HEARNSHAW +We define a cluster to be any subset P ⊂ {1, . . . , N}. Denote P c = {1, . . . , N}\P, +P ∗ = P\{1}. We will also need cluster sets, P = (P1, . . . , PM), where M ≥ 1 and +P1, . . . , PM are clusters. +First-order cluster derivatives are defined, for a non-empty cluster P, by +(1.15) +Dα +P = +� +j∈P +∂α +xj +for α ∈ N3 +0, |α| = 1. +For P = ∅, Dα +P is defined as the identity. +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, +(1.16) +Dα +P = (De1 +P )α′(De2 +P )α′′(De3 +P )α′′′ +where e1, e2, e3 are the standard unit basis vectors of R3. Let P = (P1, . . . , PM) and +α = (α1, . . . , αM), αj ∈ N3 +0, 1 ≤ j ≤ M, then we define the multicluster derivative (often +simply referred to as cluster derivative) by +(1.17) +Dα +P = Dα1 +P1 . . . DαM +PM . +It can readily be seen that cluster derivatives obey the Leibniz rule. +Throughout, the letter C refers to a positive constant whose value is unimportant but +may depend on Z, N and the eigenvalue E. +Distance function notation and elementary results. For non-empty cluster P, +define +ΣP = +� +x ∈ R3N : +� +j∈P +|xj| +� +l∈P +m∈P c +|xl − xm| = 0 +� +. +(1.18) +For P = ∅ we set ΣP := ∅. Denote Σc +P = R3N\ΣP. For each P we have ΣP ⊂ Σ where +(1.19) +Σ = +� +x ∈ R3N : +� +1≤j≤N +|xj| +� +1≤l 0, +f∞(x; r; u) := ∥∇u∥L∞(B(x,r)) + ∥u∥L∞(B(x,r)) +(1.28) +for x ∈ R3N. The ball B(x, r) is considered in R3N. Largely, this notation will be used +for u = ψ and in this case we have the notation, +f∞(x; r) := f∞(x; r; ψ). +(1.29) +A pointwise cluster derivative bound. To prove Theorem 1.1 we will state and +prove a new pointwise bound to cluster derivatives of eigenfunctions ψ, which may itself +be of independent interest. It will be shown by elliptic regularity that for all α the weak +cluster derivatives Dα +Pψ exist in the set Σc +α. It is therefore interesting to consider how +such cluster derivatives behave as the set Σα is approached. +Previously, S. Fournais and T. Ø. Sørensen have given bounds to local Lp-norms of +cluster derivatives of ψ for a single cluster P. Indeed, in [8, Proposition 1.10] 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 +(1.30) +∥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. Notice that for every x ∈ Σc +P, we have B(x, rλP(x)) ⊂ Σc +P by the +definition of λP(x). Therefore, the set ΣP is avoided when evaluating Dα +Pψ in the Lp- +norms. +The objective of the following theorem is to extend the bounds (1.30) in the case of +p = ∞ and for cluster derivatives for cluster sets P. In particular, estimates are obtained +which depend on the order of derivative for each of the respective clusters in P. In the +following, ∇ denotes the gradient operator in R3N. +Theorem 1.3. For every cluster set P = (P1, . . . , 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, +(1.31) +��Dα +P∇kψ +�� +L∞(B(x,rλα(x))) ≤ Cλα(x)1−kλP1(x)−|α1| . . . λPM(x)−|αM|f∞(x; R) +for all x ∈ Σc +α. +Furthermore, for each |α| ≥ 1 there exists a function Gα +P : Σc +α → C3N such that +(1.32) +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 +(1.33) +∥Gα +P∥L∞(B(x,rλα(x))) ≤ Cµα(x)bλP1(x)−|α1| . . . λPM(x)−|αM|f∞(x; R) +for all x ∈ Σc +α. +Remark 1.4. +(1) In the case of a single cluster and k = 0, the bound (1.31) reestab- +lishes (1.30) in the case of p = ∞, albeit with a slightly larger radius in the +L∞-norms on the right-hand side. +(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, . . . , M with αi ̸= 0. Notice that the appropriate j will depend on x. +(3) The bound in (1.33) is stronger than that of (1.31) with k = 1. 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, . . . , M with αi ̸= 0. +The proof of Theorem 1.3 will follow a similar strategy to that of [8, Proposition 1.10]. +An additional result will be required to prove (1.33). 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. +We will require an elliptic regularity result, stated below, which will be used in the +proofs. Beforehand, we clarify the precise form of definitions which we will be using. +Let Ω be open, θ ∈ (0, 1] and k = N0. 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. + +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. In addition, the space Ck,θ(Ω) is defined as all f ∈ Ck(Ω) +where [∇kf]θ,Ω is finite. This space has a norm given by +∥f∥Ck,θ(Ω) = ∥f∥Ck(Ω) + [∇kf]θ,Ω. +For open Ω ⊂ Rn we can consider the following elliptic equation, +(1.34) +Lu := −∆u + c · ∇u + du = g +for some c : Ω → Cn and d, g : Ω → C. 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). We say that a function u ∈ H1 +loc(Ω) is a weak +solution to the equation (1.34) in Ω if L(u, χ) = +� +Ω gχ dx for every χ ∈ C∞ +c (Ω). +The following theorem is a restatement of [5, Proposition 3.1] ([8, Proposition A.2] is +similar), with additional H¨older regularity which follows from the proof. +Theorem 1.5. Let x0 ∈ Rn, R > 0 and c, d, g ∈ L∞(B(x0, R)) and u ∈ H1(B(x0, R)) +be a weak solution to (1.34) 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 +(1.35) +∥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. +2. Proof of Theorem 1.3 +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. 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. The function e−Fψ will solve an elliptic equation with bounded +coefficients which behave suitably well under the action of cluster derivatives. Elliptic +regularity will then produce bounds to the cluster derivatives of e−Fψ. Such bounds can +then be used to obtain bounds to the cluster derivatives of ψ itself. +2.1. Jastrow factors. We begin by defining the function +(2.1) +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 0), such that +(2.10) +C ∥φ∥L∞(B(x,R)) ≤ ∥ψ∥L∞(B(x,R)) ≤ C′ ∥φ∥L∞(B(x,R)) +for all x ∈ R3N. +2.2. Derivatives of F. Informally, our objective is to take cluster derivatives of the +elliptic equation (2.8) and apply elliptic regularity. To do so, we require bounds to the +cluster derivatives of the coefficients present in this equation. This is the objective of the +current section. To begin, we state and prove the following preparatory lemma involving +the distances introduced in (1.21), (1.25) and (1.27). + +ONE-PARTICLE DENSITY MATRIX +9 +Lemma 2.1. For any σ = (σ1, . . . , σM) ∈ N3M +0 +we have for k = 0, 1, +(2.11) +µσ(y)k−|σ| ≤ λσ(y)kλP1(y)−|σ1| . . . λPM(y)−|σM| +for all y ∈ R3N. Furthermore, let β(1), . . . , β(n) be an arbitrary collection of multiindices +in N3M +0 +such that β(1) + · · · + β(n) = σ then +(2.12) +n +� +j=1 +λβ(j)(y) ≤ λσ(y) +for all y ∈ R3N. +Proof. The results are trivial in the case of σ = 0 therefore we assume in the following +that σ is non-zero. First, observe for all j = 1, . . . , M, +µσ(y)−|σj| ≤ λPj(y)−|σj| +µσ(y)1−|σj| ≤ λPj(y)1−|σj| +if σj ̸= 0 +by the definition of µσ. Now perform the following trivial expansion of the product, +µσ(y)−|σ| = µσ(y)−|σ1| . . . µσ(y)−|σM| +≤ λP1(y)−|σ1| . . . λPM(y)−|σM| +which proves (2.11) for k = 0. For k = 1, consider that for each y we can find l = +1, . . . , M such that λσ(y) = λPl(y) and σl ̸= 0. Then, +µσ(y)1−|σ| = µσ(y)−|σ1| . . . µσ(y)1−|σl| . . . µσ(y)−|σM| +≤ λP1(y)−|σ1| . . . λPl(y)1−|σl| . . . λPM(y)−|σM| += λσ(y)λP1(y)−|σ1| . . . λPM(y)−|σM| +as required. +. Finally we prove (2.12). As above, take arbitrary y and a corresponding l such that +λσ(y) = λPl(y) with σl ̸= 0. For each 1 ≤ j ≤ n we denote the N3 +0-components of β(j) as +β(j) = (β(j) +1 , . . . , β(j) +M ). We know β(1)+· · ·+β(n) = σ so in particular, β(1) +l ++· · ·+β(n) +l += σl. +Since σl ̸= 0 there exists at least one 1 ≤ r ≤ n such that β(r) +l +̸= 0. Hence by the definition +of λβ(r), +λβ(r)(y) = min{λPs(y) : β(r) +s +̸= 0, s = 1, . . . , M} ≤ λPl(y) = λσ(y). +The remaining factors λβ(j)(y), for j ̸= r, can each be bounded above by one. +□ +The following lemma will be useful in proving results about taking cluster derivatives +of F, as defined in (2.1). Later, we will apply it using f as the function |x| for x ∈ R3, +or derivatives thereof. + +10 +PETER HEARNSHAW +Lemma 2.2. Let f ∈ C∞(R3\{0}) and k ∈ N0 be such that for each σ ∈ N3 +0 there exists +C such that +(2.13) +|∂σf(x)| ≤ C|x|k−|σ| for all x ̸= 0. +Then for any α ̸= 0 with |α| ≥ k there exists some new C such that for any l, m = +1, . . . , 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. +Proof. Take any j = 1, . . . , M with αj ̸= 0, then we have D +αj +Pj(f(xl)) ≡ 0 for each l ∈ P c +j . +Therefore, for Dα +P(f(xl)) to not be identically zero we require that l ∈ Pj for each j with +αj ̸= 0. For such l we have xl ̸= 0 since x ∈ Σc +α, and +|xl| ≥ dPj(x) +for each j with αj ̸= 0 by (1.20). Therefore, for constant C in (2.13), we have +|Dα +P(f(xl))| = |∂α1+···+αMf(xl)| ≤ C|xl|k−|α| ≤ Cqα(x)k−|α| +because |α| ≥ k. Similarly, for each j = 1, . . . , M, with αj ̸= 0 we have D +αj +Pj(f(xl−xm)) ≡ +0 if either l, m ∈ Pj or l, m ∈ P c +j . 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) +� +. +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 (1.20). 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. +□ +The following lemma provides pointwise bounds to cluster derivatives of functions +involving F. +Lemma 2.3. For any cluster set P and any |σ| ≥ 1 there exists C, which depends on +σ, such that for k = 0, 1, +(2.14) +��Dσ +P∇kF(y) +��, +��Dσ +P∇k(eF)(y) +�� ≤ Cλσ(y)1−kλP1(y)−|σ1| . . . λPM(y)−|σM| +(2.15) +��Dσ +P|∇F(y)|2�� ≤ CλP1(y)−|σ1| . . . λPM(y)−|σM| +for all y ∈ Σc +σ. The bound to the first object in (2.14) also holds when F is replaced by +Fc. + +ONE-PARTICLE DENSITY MATRIX +11 +Proof. Let τ be the function defined as τ(x) = |x| for x ∈ R3. Then, by definition (2.2) +we can write +(2.16) +Fc(y) = −Z +2 +� +1≤j≤N +τ(yj) + 1 +4 +� +1≤l 0 we can define the following two cutoff factors ζt = ζt(z) and +θt = θt(z) by +(3.2) +ζt(z) = χ +�4N|z| +t +� +, +θt(z) = 1 − ζt(z) +for z ∈ R3. We have the following support criteria for cutoff factors. 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. +Let 0 < δ < (4N)−1ǫ (the use of (4N)−1 is explained in the following lemma). We +define a biscaled cutoff, which depends on δ and ǫ as parameters, as a function Φ = +Φδ,ǫ(x, y, ˆx) defined by +(3.3) +Φδ,ǫ(x, y, ˆx) = +� +2≤j≤N +g(1) +j (x − xj) +� +2≤j≤N +g(2) +j (y − xj) +� +2≤k 0, +1t(z) = 1{(4N)−1t<|z|<(2N)−1t}(z) +(3.4) +1′ +t(z) = 1{(4N)−1t<|z|<1}(z) +(3.5) +for z ∈ R3. 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 (4N)−1ǫ ≥ δ. For such z we therefore have θδ(z) = 1, by the definition +of θδ, which proves (3.7). For (3.8), if we also have ζδ(z) ̸= 0, then |z| < (2N)−1δ by the +support criteria for ζδ, giving a contradiction. For (3.9) we need only consider z such +that ζδ(z) ̸= 0, in which case |z| < (2N)−1δ. This gives 4N|z|ǫ−1 < (2N)−1 and hence, +by definition, ζǫ(z) = 1. +. For any A, B ⊂ {2, . . . , N} with A ∩ B = ∅ we define +τA,B(x) = +� +j∈A +ζδ(x1 − xj) +� +j∈B +(θδζǫ)(x1 − xj) +� +j∈{2,...,N}\(A∪B) +θǫ(x1 − xj) +and therefore +� +A⊂{2,...,N} +B⊂{2,...,N}\A +τA,B(x) = +� +2≤j≤N +� +ζδ(x1 − xj) + (θδζǫ)(x1 − xj) + θǫ(x1 − xj) +� += 1 +for all x ∈ R3N. Let Ξ = {(j, k) : 2 ≤ k < l ≤ N}. 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 δ/2 by the +reverse triangle inequality. The case of k ∈ S∗ is analogous. + +22 +PETER HEARNSHAW +. Now let k ∈ Q∗. First, we consider the case where either g(1) +k +̸= θǫ or g(2) +k +̸= θǫ, or both. +Without loss, assume g(1) +k +̸= θǫ. Then either g(1) +k += ζδ or g(1) +k += θδζǫ. 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. Now, suppose that g(1) +k += g(2) +k += θǫ. +Then there exist pairwise distinct j1, . . . , js ∈ {2, . . . , N} with 1 ≤ s ≤ N − 2 such that +fj1,j2, fj2,j3, . . . , fjs,k ̸= θǫ and either g(1) +j1 ̸= θǫ or g(2) +j1 ̸= θǫ. 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. It’s also clear that by support +criteria, |xj1 − xj2|, . . . , |xjs − xk| ≤ (2N)−1ǫ. 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. +□ +3.4. Factorisation of biscaled cutoffs. Let Φ be given by (3.3). We can define a +partial product of Φ as a function of the form +(3.14) +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, . . . , N}, R1 ⊂ {(k, l) : 2 ≤ k < l ≤ N}. +We now define classes of partial products of Φ which corresponding to a cluster. Let +T be an arbitrary cluster with 1 ∈ T. +Φ(x, y, ˆx; T) = +� +j∈T ∗ +g(1) +j (x − xj) +� +j∈T ∗ +g(2) +j (y − xj) +� +k,l∈T ∗ +k 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. +Recall the function 1t was defined in (3.4). +Lemma 3.5. For any σ ∈ N3 +0 with |σ| ≥ 1 and any t > 0 there exists C, depending on +σ but independent of t, such that +(3.22) +|∂σζt(z)|, |∂σθt(z)| ≤ Ct−|σ|1t(z) +for all z ∈ R3. + +24 +PETER HEARNSHAW +Proof. Without loss we consider the case of θt, the case of ζt being similar. +In the +following, χ(j) refers to the j-th (univariate) derivative of the function χ defined in (3.1). +Now, since |σ| ≥ 1 the chain rule shows that ∂σθt(z) can be written as a sum of terms +of the form +(3.23) +�4N +t +�m +χ(m)�4N|z| +t +� +∂σ1 +z |z| . . . ∂σm +z |z| +where 1 ≤ m ≤ |σ|, and σ1, . . . , σm ∈ N3 +0 are non-zero multiindices obeying +σ1 + · · · + σm = σ. +Since m ≥ 1 we have that if χ(m)(s) ̸= 0 then s ∈ (1, 2), and therefore for any term +(3.23) to be non-zero we require that +(3.24) +(4N)−1t < |z| < (2N)−1t. +By the remark preceeding the current lemma, there exists C, dependent on σ1, . . . , σm, +such that +∂σ1 +z |z| . . . ∂σm +z |z| ≤ C|z|m−|σ| ≤ C(4N)|σ|−mtm−|σ|, +using (3.24). Therefore, the terms (3.23) can readily be bounded to give the desired +result. +□ +We now give bounds for the cluster derivatives (3.20)-(3.21) acting on cutoffs. +Lemma 3.6. Let Φ be any biscaled cutoff and let Q = Q(Φ). Then +Dα +x,y,QΦ( · ; Q) ≡ 0 +(3.25) +for all α ∈ N3 +0 with |α| ≥ 1. +Proof. By the chain rule, each function in the product (3.15) for Φ( · ; Q) has zero deriv- +ative upon action of Dα +x,y,Q. +□ +Recall that the notion of partial products was defined in (3.14) and the function Mt +for t > 0 was defined in (3.6). +Lemma 3.7. Let δ ≤ (4N)−1ǫ and Φ = Φδ,ǫ be a biscaled cutoff. Let Q = Q(Φ). 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( · ; Q, Qc) +C +� +ǫ−|α| + δ−|α|Mδ(x, y, ˆx) +� +otherwise. +(3.26) +Proof. Lemma 3.5 gives bounds for the partial derivatives of the function ζδ, θδ, ζǫ, θǫ. +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), +(3.27) + +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 3.5 and the definition (3.4). +. Now, to evaluate ∂αΦ′ we apply the Leibniz rule to the product (3.14) using (3.27) +where appropriate. Differentiated cutoff factors are bounded by (3.22) and any remaining +undifferentiated cutoff factors are bounded above by 1. If Φ′ = Φ( · ; Q, Qc), all cutoff +factors are of the form θǫ by Lemma 3.4 and therefore we need only use the bounds in +(3.22) with t = ǫ. +□ +The derivative Dx,y,Q acting on Φ is special in that it contributes only powers of ǫ (and +not δ) to the bounds. This is shown in the next lemma. +Lemma 3.8. Let δ ≤ (4N)−1ǫ and Φ = Φδ,ǫ be a biscaled cutoff. Let Q = Q(Φ). 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. +Proof. First, set Φ′ = Φ( · ; Q) Φ( · ; Qc) and Φ′′ = Φ( · ; Q, Qc). We then have +Dα +x,y,QΦ = Φ′ Dα +x,y,QΦ′′ +which follows from Lemmas 3.3 and 3.6 and that Φ( · ; Qc) is not dependent on variables +involved in the Dα +x,y,Q-derivative. By the definition (3.21), 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|. Now, by the Leibniz rule and Lemma 3.7 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. +□ +4. Proof of Theorem 1.1 +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. For the density matrix we then estimate the resulting integrals involving cluster +derivatives of ψ using the pointwise bounds of Theorem 1.3. + +26 +PETER HEARNSHAW +Although Theorem 1.1 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. Indeed, +we define new variables +u = (x + y)/2, +(4.1) +v = (x − y)/2, +(4.2) +and consider the ∂α +u ∂β +v -derivatives of the density matrix. 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. +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. +Consider derivatives of the form ∂α +x ∂β +y γ(x, y) where |α|+|β| = 2. As discussed in [5], the +case where |α| = |β| = 1 (the mixed derivatives) is particularly well-behaved in contrast +to the other cases. The reason behind this is that when |α| = |β| = 1, differentiation +under the integral leads to both ψ factors being differentiated exactly once. 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. +This is used in the proof in the following way. In (4.34) 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. 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. +This method only works for two v-derivatives, and the +strategy of using cluster derivatives, as described above, must be used in conjunction. +The difficulties encountered in the fifth derivative of the density matrix are described in +a later section. +4.1. Density matrix notation. The proof will require auxiliary functions related to +the density matrix which we introduce now. For l, m ∈ N3 +0 with |l|, |m| ≤ 1 define, +(4.3) +γl,m(x, y) = +� +R3N−3 ∂l +xψ(x, ˆx)∂m +y ψ(y, ˆx) dˆx. +In this notation, it is clear that γ = γ0,0. For any biscaled cutoff Φ, defined in (3.3), we +set +γl,m(x, y; Φ) = +� +R3N−3 ∂l +xψ(x, ˆx)∂m +y ψ(y, ˆx)Φ(x, y, ˆx) dˆx, +and define γ( · ; Φ) = γ0,0( · ; Φ). We will consider the above functions in the variables +(4.1) and (4.2). It is then natural to define for all u, v ∈ R3, +˜γl,m(u, v) = γl,m(u + v, u − v), +(4.4) +˜γl,m(u, v; Φ) = γl,m(u + v, u − v; Φ). +(4.5) + +ONE-PARTICLE DENSITY MATRIX +27 +4.2. Integrals involving f∞. The following proposition is a restatement of [5, Lemma +5.1] and is proven in that paper. Notice that the function Mt was defined in (3.6) and +has a slightly different form to the corresponding function used in the paper. +Proposition 4.1. Given R > 0, there exists C such that +(4.6) +� +R3N−3 f∞(x, ˆx; R)f∞(y, ˆx; R) dˆx ≤ C ∥ρ∥1/2 +L1(B(x,2R)) ∥ρ∥1/2 +L1(B(y,2R)) +for all x, y ∈ R3. In addition, given G ∈ L1(R3) there exists C, independent of G, such +that +(4.7) +� +R3N−3 +� +|G(xj − xk)| + |G(z − xk)| + |G(xj)| +� +f∞(x, ˆx; R)f∞(y, ˆx; 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, . . . , N, j ̸= k. In particular, for any t > 0 there exists +C, independent of t, such that +(4.8) +� +R3N−3 Mt(x, y, ˆx)f∞(x, ˆx; R)f∞(y, ˆx; R) dˆx ≤ Ct3 ∥ρ∥1/2 +L1(B(x,2R)) ∥ρ∥1/2 +L1(B(y,2R)) +for all x, y ∈ R3. +We introduce the following quantities based on the bounded distances (1.21). 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)}, +(4.9) +π(x, y, ˆx) = min{λQ(x, ˆx), λQ(y, ˆx)}. +(4.10) +Later, we will see that these quantities appear to negative powers when we apply Theorem +1.3 to clusters derivatives of ψ involving the clusters P, S and Q. The next lemma will +give conditions for when these quantities can be bounded away from zero on the support +of Φ. +Beforehand, we consider an alternative formulae for (4.10) and (4.9). +Recall that +1 ∈ P, S, Q by definition. Using (1.20) we find that +(4.11) +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}. +In the case of P ∗ ∩ S∗ = ∅ we similarly find that +(4.12) +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)}. + +28 +PETER HEARNSHAW +Lemma 4.2. Let δ ≤ (4N)−1ǫ, and ǫ ≤ 1 and let Φ = Φδ,ǫ be an arbitrary biscaled +cutoff. Let x, y ∈ R3 be such that δ ≤ |x − y| ≤ 2δ, and |x|, |y| ≥ ǫ. Then there exists a +constant C, dependent only on N, such that +π(x, y, ˆx) ��� Cǫ +(4.13) +λ(x, y, ˆx) ≥ Cδ +(4.14) +whenever Φ(x, y, ˆx) ̸= 0. 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 +(4.15) +� +supp Φ(x,y, · ) +λ(x, y, ˆx)−bf∞(x, ˆx; R)f∞(y, ˆx; 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. +The following corollary will be useful later. +Corollary 4.3. There exists C, depending on R but independent of δ and ǫ, such that +(4.16) +2 +� +r=1 +� +R3N−3 λ(x, y, ˆx)−2f∞(x, ˆx; R)f∞(y, ˆx; 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. +Proof of Corollary 4.3. When r = 1 the bound for the integral is immediate. +Using +Lemma 3.8 and (4.14), the integral in (4.16) 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; R)f∞(y, ˆx; R) dˆx +The required bound follows from (4.15) of Lemma 4.2, (4.8) of Proposition 4.1 and that +ǫ < 1. +□ +Proof of Lemma 4.2. By Lemma 3.2 we need only consider Φ with P ∗ ∩ S∗ = ∅. By the +definition of the cluster Q, if j ∈ Q∗ and k ∈ Qc then fjk = θǫ in the formula (3.3) +defining Φ. Similarly, if k ∈ Qc then g(1) +k += g(2) +k += θǫ. Using the support criteria of θǫ we +therefore get +|x − xk|, |y − xk|, |xj − xk| ≥ (4N)−1ǫ +j ∈ Q∗, k ∈ Qc. +(4.17) + +ONE-PARTICLE DENSITY MATRIX +29 +In addition, if j ∈ Q∗ we get +(4.18) +|xj| ≥ |x| − |x − xj| ≥ ǫ/2 +j ∈ Q∗ +since for such j we have |x − xj| ≤ ǫ/2 by Lemma 3.2. The lower bound (4.13) then +follows from the formula (4.11) for π(x, y, ˆx). +. In a similar way we consider the clusters P and S. Indeed, let j ∈ P ∗, k ∈ P c or j ∈ S∗, +k ∈ Sc, then fjk ̸= ζδ. Similarly, if k ∈ P c then g(1) +k +̸= ζδ, and if k ∈ Sc then g(2) +k +̸= ζδ. +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. Therefore by the support criteria of these factors we +obtain the following inequalities, +|x − xk|, |y − xl| ≥ (4N)−1δ +k ∈ P c, l ∈ Sc +(4.19) +|xj − xk| ≥ (4N)−1δ +j ∈ P ∗, k ∈ P c or j ∈ S∗, k ∈ Sc. +(4.20) +We also have (4.18) for j ∈ P ∗ ∪S∗ since P, S ⊂ Q. The lower bound (4.14) then follows +from the formula (4.12) for λ(x, y, ˆx). +. We recall the function 1′ +δ was defined in (3.5). By the formula (4.12) we can use (4.18) +to obtain some C, depending on b, such that +(4.21) +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 (3.5) can be included because 1 lies in +the minimum (4.12) and the lower bounds follow from (4.18)-(4.20). The bound (4.15) +then follows from the above inequality along with both (4.6) and (4.7) of Proposition +4.1, where we choose G to be the function G(z) = 1′ +δ(z)|z|−b for z ∈ R3. +□ +4.3. Differentiating the density matrix - some required bounds. We collect cer- +tain results which will be used throughout this section. Let δ ≤ (4N)−1ǫ and ǫ ≤ 1 and +suppose Φ = Φδ,ǫ is a biscaled cutoff as defined in (3.3). As usual, we denote Q = Q(Φ), +P = P(Φ) and S = S(Φ). Let η ∈ N3 +0 and ν = (ν1, ν2) ∈ N6 +0 be arbitrary. Firstly, we +define +(4.22) +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. +By Lemma 3.8 and the definition of cluster derivatives (1.16) it can be shown that for +each η and ν there exists C, independent of δ and ǫ, such that +(4.23) +|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. + +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. +Then, as a consequence of Lemma 4.2 we have both π(x, y, ˆx) and λ(x, y, ˆx) are +positive. Therefore, by the definitions (4.9), (4.10) and (1.20), (1.21), +(x, ˆx) ∈ Σc +Q ∩ Σc +P ∩ Σc +S∗ +and +(y, ˆx) ∈ Σc +Q ∩ Σc +P ∗ ∩ Σc +S. +(4.24) +This allows us to apply Theorem 1.3. 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; R), +(4.25) +��Dη +QDν +{P ∗,S}∇kψ(y, ˆx) +�� ≤ Cπ(x, y, ˆx)−|η|λ(x, y, ˆx)−|ν|f∞(y, ˆx; R). +(4.26) +For convenience, we choose a bound which holds for both values of k. The functions λ +and π are defined in (4.9) and (4.10) respectively. +Now, let |ν| ≥ 1. By Theorem 1.3 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) +� +(4.27) +Dν +{P ∗,S}∇ψ(y, ˆx) = Gν +{P ∗,S}(y, ˆx) + ψ(y, ˆx) +� +Dν +{P ∗,S}∇Fc(y, ˆx) +� +(4.28) +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; R), +(4.29) +��Gν +{P ∗,S}(y, ˆx) +�� ≤ Cλ(x, y, ˆx)1/2−|ν|f∞(y, ˆx; R), +(4.30) +for some C, dependent on ν but independent of the choice of x, y and ˆx. +By Lemma 2.3 and (2.2), 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|. +(4.31) +4.4. Differentiating the density matrix - first and second derivatives. In the +following, let l, m ∈ N3 +0 obey |l| = |m| = 1. In standard notation, by ∂l +x1ψ we mean the +l-partial derivative in the first R3 component of ψ. 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 +(4.32) += ˜γl,0(u, v) + ˜γ0,l(u, v), +and similarly, +∂l +v˜γ(u, v) = ˜γl,0(u, v) − ˜γ0,l(u, v) +(4.33) + +ONE-PARTICLE DENSITY MATRIX +31 +for all u, v ∈ R3. The above equalities are used to obtain a formula relating second order +u-derivatives to second order v-derivatives of ˜γ. 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) +(4.34) +where the final equality is obtained by differentiation under the integral, as in (4.32). +4.5. Differentiating the density matrix - general derivatives. Partial derivatives +of γ are written as linear combinations of integrals involving cluster derivatives of ψ. +One such integral is bounded in the following lemma. Since it is more involved than the +other such integrals, the proof is postponed until later in the section. +Lemma 4.4. Let δ ≤ (4N)−1ǫ and ǫ ≤ 1 and let Φ = Φδ,ǫ be an arbitrary biscaled cutoff. +Let P = P(Φ) and S = S(Φ) obey P ∗ ∩ S∗ = ∅. 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 +(4.35) +��� +� +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. The constant C depends on R but is independent +of δ, ǫ. +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. We use notation +of cluster derivatives of Φ from (4.22). +Lemma 4.5. Let δ ≤ (4N)−1ǫ and ǫ ≤ 1 and Φ = Φδ,ǫ be an arbitrary biscaled cutoff. +Let η, µ ∈ N3 +0 be arbitrary and l, m ∈ N3 +0 be such that |l|, |m| ≤ 1. +(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; Φ) is equal to a linear combination of integrals of the form +(4.36) +� +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|. +(ii) Furthermore, suppose either +• |µ| ≤ 2, or +• |µ| = 3, η = 0 and |l| = |m| = 1. + +32 +PETER HEARNSHAW +Then for all R > 0 we have some C0 such that +(4.37) +|∂η +u∂µ +v ˜γl,m(u, v; 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. The constant C0 depends on R but +is independent of δ, ǫ. +Proof of i). By Lemma 3.2, we need only consider Φ such that P ∗ ∩ S∗ = ∅. For each +choice of u and v we define a u- and v-dependent change of variables for the integral +˜γlm(u, v; Φ) defined in (4.5). +To start, we define two vectors ˆa = (a2, . . . , aN), ˆb = +(b2, . . . , bN) ∈ R3N−3 by +ak = +� +u +if k ∈ Q∗ +0 +if k ∈ Qc +bk = + + + + + +v +if k ∈ P ∗ +−v +if k ∈ S∗ +0 +if k ∈ (P ∪ S)c, +(4.38) +and define ˆωu,v = ˆa + ˆb. We then apply a translational change of variables which allows +us to write +(4.39) +˜γlm(u, v; 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. +We will then apply differentiation under the integral. Beforehand, we show how such +derivatives will act on each function within the integrand. 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). +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) +� +. +where cτ,ν = (−1)|ν|−|τ|�ν +τ +� +. In a similar manner, for the cutoff we use the definitions +(3.20)-(3.21) and (4.22) 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). +By differentiating (4.39) under the integral, applying the Leibniz rule and reversing the +change of variables, we find that ∂η +u∂µ +v ˜γlm(u, v; Φ) is a linear combination of terms of the +required form. + +ONE-PARTICLE DENSITY MATRIX +33 +Proof of ii). By part (i) it suffices to prove the required bound for integrals of the form +(4.36) with |χ| = |η| and |α| + |β| + |σ| = |µ|. Rewriting such integrals in the variables +x = u + v and y = u − v we get +(4.40) +� +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. +Notice that the variables x and y must obey δ ≤ |x − y| ≤ 2δ and |x|, |y| ≥ ǫ. Firstly, +we bound the above integral in the case where |α| + |β| ≤ 2 and |α| + |β| + |σ| ≤ 3 for +any |l|, |m| ≤ 1. By (4.23), (4.25), (4.26) followed by (4.13) of Lemma 4.2 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; R)f∞(y, ˆx; R) dˆx. +Selective use of (4.14) of Lemma 4.2 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; R)f∞(y, ˆx; R) dˆx. +We can use (4.15) of Lemma 4.2, using that |α| + |β| ≤ 2 by assumption, and (4.8) of +Proposition 4.1 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. After a return to u, v- +variables, this proves the bound for integrals (4.36) in the case where |α| + |β| ≤ 2 +and |α| + |β| + |σ| ≤ 3. It remains to bound (4.40) in the case where |α| + |β| = 3, +|σ| = |χ| = 0 and |l| = |m| = 1. This follows directly from Lemma 4.4. +□ +Lemma 4.6. Take any η, µ, l, m ∈ N3 +0 as in Lemma 4.5(ii). Then for all R > 0 we have +C such that +(4.41) |∂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|}. +Proof. Firstly, by Lemma 3.1 there exists a finite collection of biscaled cutoffs, Φ(j), +j = 1, . . . , J, such that +(4.42) +˜γl,m = +J +� +j=1 +˜γl,m +� +· ; Φ(j) +δ,ǫ +� +holds for all choices of 0 < δ ≤ (4N)−1ǫ. + +34 +PETER HEARNSHAW +Let C0 be the constant from Lemma 4.5 such that (4.40) holds for each Φ(j), j = +1, . . . , J. 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|}. Then by (4.42), +|∂η +u∂µ +v ˜γl,m(u0, v0)| ≤ +J +� +j=1 +��∂η +u∂µ +v ˜γl,m +� +u0, v0; 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)) . +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. +□ +Proposition 4.7. For all α, β ∈ N3 +0 with |α| + |β| = 5 and all R > 0 there exists C, +depending on R, such that +(4.43) +|∂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|}. +Proof. First, we consider the case where |β| ≤ 3. We use (4.32) and (4.33) for when |β| ≤ +2 and |β| = 3 respectively. The bound then follows from Lemma 4.6 with |η| + |µ| = 4 +and |µ| ≤ 2. Now, consider 4 ≤ |β| ≤ 5. Take l, m ∈ N3 +0, |l| = |m| = 1 be such that +l + m ≤ β. Then by (4.34), +∂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) +� +. +The first term on the right-hand side has |β| − |l| − |m| ≤ 3 hence the required bound +follows from the previous step. The remaining terms can be bounded using Lemma 4.6 +□ +The proof of our main theorem is an immediate consequence of this proposition. +Proof of Theorem 1.1. The proof follows from Proposition 4.7 with u = (x + y)/2 and +v = (x − y)/2, along with the definition (4.4). +□ +4.6. Proof of Lemma 4.4. To prove Lemma 4.4, we will examine the cluster derivatives +of ψ present in (4.35) more closely. In particular, Theorem 1.3 allows us to write such +derivatives in terms of derivatives of Fc and a function, Gα +P, of higher regularity near +certain singularities. Sign cancellation allows uniform boundedness of the integral (4.35) +as x and y approach each other, and is more easily handled via derivatives of Fc rather +than those of ψ itself. Indeed, due to the simple formula definining Fc, it is possible +to characterise all its cluster derivatives explicitly. A series of steps, mostly involving +integration by parts, will complete the proof. +To begin, we use definition (2.2) to write +∇xFc(x, ˆx) = −Z +2 ∇x|x| + 1 +4 +� +2≤j≤N +∇x|x − xj| +(4.44) + +ONE-PARTICLE DENSITY MATRIX +35 +with the formula also holding when x is replaced by y. Let P and S be arbitrary clusters +with 1 ∈ P, S and P ∗ ∩ S∗ = ∅. Let α = (α1, α2) ∈ N6 +0, β = (β1, β2) ∈ N6 +0 and let +l, m ∈ N3 +0 obey |l| = |m| = 1. Then, +Dα +{P,S∗}∂l +x|x| = +� +∂α1+l +x +|x| +if α2 = 0 +0 +if |α2| ≥ 1, +(4.45) +Dβ +{P ∗,S}∂m +y |y| = +� +∂β2+m +y +|y| +if β1 = 0 +0 +if |β1| ≥ 1. +(4.46) +In the following, the cluster derivatives in (4.47) are understood to act with respect to +the ordered variables (x, x2, . . . , xN) and the cluster derivatives in (4.48) are understood +to act with respect to the ordered variables (y, x2, . . . , xN). 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. Now assume |α|, |β| ≥ 1, then +Dα +{P,S∗}∂l +x|x − xj| = + + + + + + + + + +(−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 ∗ +(4.47) +Dβ +{P ∗,S}∂m +y |y − xj| = + + + + + + + + + +(−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∗. +(4.48) +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. +We will often use the following elementary fact. For each z0 ∈ R3 and η ∈ N3 +0 there +exists C, dependent on η but independent of z0, such that +(4.49) +��∂η +z |z0 − z| +�� ≤ C|z0 − z|1−|η| +for all z ̸= z0. +Therefore, by (4.12) 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| +(4.50) +for all 2 ≤ j, k ≤ N if |η| = 1, and all j ∈ P c, k ∈ Sc if |η| ≥ 2. Here, λ(x, y, ˆx) is defined +in (4.9) using the clusters P and S. +Lemma 4.8. Let P = P(Φ) and S = S(Φ) obey P ∗ ∩ S∗ = ∅, and take any R > 0. +Consider |α| + |β| = 3 and |l| = |m| = 1. For |α|, |β| ≥ 1, the integral +(4.51) +� +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 (4.35). 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, +(4.52) +� +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, +(4.53) +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, +(4.54) +� +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 +(4.55) +can each be bounded as in (4.35). +Proof. We first prove the bound for (4.51). Use of (4.44) and (4.45)-(4.48) 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. +Derivatives of |x| and |y| are bounded using (4.49) and |x|, |y| ≥ ǫ. Now we bound each +integral in absolute value. The first, second and third integrals are readily bounded by +(4.50) and Lemma 4.2. Finally, for the fourth integral we use (4.59) of Lemma 4.9. +Now suppose |α| = 3. +First, we prove the bound for the integral (4.52). +Recall +F = Fc − Fs. Using this, along with (4.44) and (4.45)-(4.48) we can expand the integral + +ONE-PARTICLE DENSITY MATRIX +37 +as linear combination of the following terms, where j ∈ P c and k ∈ {2, . . . , N}, +∂α1+l +x +|x|∂m +y |y| +� +R3N−3 ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +if α2 = 0, +∂α1+l +x +|x| +� +R3N−3 ∂m +xk|y − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +if α2 = 0, +∂α1+l +x +|x| +� +R3N−3 ∂m +xkFs(y, ˆx)ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +if α2 = 0, +∂m +y |y| +� +R3N−3 ∂α1+α2+l +xj +|x − xj|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx, +� +R3N−3 ∂α1+α2+l +xj +|x − xj|∂m +xk|y − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +� +R3N−3 ∂α1+α2+l +xj +|x − xj|∂m +xkFs(y, ˆx)ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +Derivatives of |x| and |y| are bounded using (4.49) and |x|, |y| ≥ ǫ. +The first three +integrals above are then readily bounded using (4.6) of Proposition 4.1 and that ∇Fs ∈ +L∞(R3N). For the fourth and sixth integral we use (4.57) of Lemma 4.9 with χ ≡ 1 +and χ(x, y, ˆx) = ∂m +xkFs(y, ˆx) respectively. Finally, for the fifth integral we use the same +lemma, specifically (4.59). This proves (4.52). The proof of (4.54) is similar in the case +where |β| = 3. +Next, using (4.44), (4.45) and (4.47) we can rewrite the integral (4.53) 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. +Derivatives of |x| are bounded using (4.49) and |x| ≥ ǫ. The first integral is then bounded +by (2.6), (2.9), (2.10) and finally (4.6) of Proposition 4.1. The second integral is bounded +immediately from (4.69) of Lemma 4.10. This proves (4.53). The proof of (4.55) is similar +in the case where |β| = 3. +□ + +38 +PETER HEARNSHAW +Proof of Lemma 4.4. To prove the required inequality, first consider the case where +|α|, |β| ≥ 1. We can then use both (4.27) and (4.28) 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. We start with the final integral on the right-hand side +which is just (4.51) of Lemma 4.8. Next, by using (4.25)-(4.26) and (4.29)-(4.31), the +first and second integrals on the right-hand side can be bounded in absolute value by +some constant multiplied by +(4.56) +� +R3N−3 λ(x, y, ˆx)−5/2f∞(x, ˆx; R)f∞(y, ˆx; 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 (4.15) of Lemma 4.2. This proves the +required bound when |α|, |β| ≥ 1. +Now we consider the case where |α| = 3, and hence β = 0. We use that ∇ψ = +ψ∇F + eF∇φ and (4.27) 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. +As before, the first integral on the right-hand side can be bounded in absolute value by +some constant multiplying (4.56). The second and third integrals on the right-hand side +are just (4.52) and (4.53) respectively. The proof of the |β| = 3 case is similar to the +|α| = 3 case. +□ +We now prove two lemmas which were used to prove Lemma 4.8. +Lemma 4.9. Let P = P(Φ) and S = S(Φ) obey P ∗ ∩ S∗ = ∅, and take any R > 0. + +ONE-PARTICLE DENSITY MATRIX +39 +(i) Let |η| = 4. Let χ = χ(x, y, ˆx) ∈ C∞(R3N+3) such that χ, ∇χ ∈ L∞(R3N+3) (for +example χ ≡ 1 may be chosen). Then, for all j ∈ P c, the integral +(4.57) +� +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 +(4.58) +� +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 (4.35). +(ii) Let |α| + |β| = 3 and |l| = |m| = 1. 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. +(4.59) +can be bounded as in (4.35). +Proof of i). Take any j ∈ P c. We bound (4.57). 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) +� +. +The functions χ and Φ are both smooth so are readily included in such a product rule. +Take any multiindex τ ≤ η with |τ| = 1. Using integration by parts we get that (4.57) +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 +(4.60) +which, using (4.50), can be bounded in absolute value by some constant, depending on χ, +multipling the expression (4.16) which is bounded in Corollary 4.3. The proof of (4.58) +is similar. +Proof of ii). We first prove the case where j ̸= k, where integration by parts is particularly +simple. Suppose |α| ≥ 1. 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 +(4.61) +which can then be bounded using Corollary 4.3. The case of |β| ≥ 1 is similar except we +apply integration by parts on the ∂m +xk-derivative. This completes the proof where j ̸= k. +For the remainder of the proof we consider the case where k = j. Suppose, first, that +k ∈ (S ∪ P)c. To simplify calculations we write η = α + l and µ = β + m. Notice that + +40 +PETER HEARNSHAW +|η|, |µ| ≥ 1. Therefore we can find some multiindex µ1 ≤ µ with |µ1| = 1. 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. +(4.62) +We leave untouched the second integral above. 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 (4.62). We retain the term where the derivative falls +on ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx). 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|. +Through this process, we obtain the following formula for Ik,k. Let T = |µ|. Then write +µ = �T +i=1 µi for some collection |µi| = 1, where 1 ≤ i ≤ T. Furthermore, define +µj = + + + + + +0 +if j = T +µT +if j = T − 1 +µj+1 + · · · + µT +if j ≤ T − 2. +Then, +(4.63) +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+µj +xk |y − xk|∂µj +xk +� +ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) +� +dˆx +Using (4.50), for each 1 ≤ j ≤ T − 1 we can bound |I(j) +k,k| by some constant multiplying +the expression (4.16), which is bounded in Corollary 4.3. +It remains to bound I(T) +k,k , +along with the first integral in the formula (4.63). Starting with the latter, we begin by +expanding |y − xk| = |x − xk| + +� +|y − xk| − |x − xk| +� +and noticing that +(4.64) +��|y − xk| − |x − xk| +�� ≤ |x − y| ≤ 2δ. +To bound the first integral in (4.63), 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, +(4.65) +� +R3N−3 ∂η+µ +xk |x − xk||x − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx. +(4.66) + +ONE-PARTICLE DENSITY MATRIX +41 +We start with the first of these two integrals. Since k ∈ (P ∪ S)c we can use (4.50) to +show that (4.65) is bounded by some constant multiplied by +(4.67) +δ +� +R3N−3 λ(x, y, ˆx)−4f∞(x, ˆx; R)f∞(y, ˆx; 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 4.2. We can then use the simplification δ(ǫ−4 +δ−1) ≤ +Cǫ−3 for some new constant C. Before looking at (4.66), we next bound I(T) +k,k . By (4.64) +we get +��I(T) +k,k +�� ≤ 2δ +� +R3N−3 +��∂η+µj = + + + + + +0 +if j = 5 +σ5 +if j = 4 +σj+1 + · · · + σ5 +if 1 ≤ j ≤ 3. +We now apply the same method used above, that is, we transfer successive first order +derivatives via integration by parts. To begin, we apply integration by parts to transfer +∂σ1 +xk from ∂σ +xk|x − xk|. We leave as a remainder the term where the derivative falls on +ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx). 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. Since |σ| = 5 is odd, the result after this procedure has occured five times is +that (4.66) is equal to minus the same integral plus remainder terms. 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 0. Let +η, l, m ∈ N3 +0 obey |η| = 4 and |l| = |m| = 1. 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 +(4.69) +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 +(4.70) +can be bounded as in (4.35). + +ONE-PARTICLE DENSITY MATRIX +43 +Proof. We prove the bound for (4.69). The case of (4.70) is similar. First, take some +function χ ∈ C∞ +c (R), 0 ≤ χ ≤ 1, with +χ(t) = +� +1 +if |t| ≤ 1 +0 +if |t| ≥ 2. +Furthermore, define χR(t) = χ(t/R) for all t ∈ R. +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 +(4.71) +� +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. +(4.72) +First, we see that when χR(|x − xj|) ̸= 1 we have |x − xj| > R. This, along with (2.6) +and (4.49) gives some constant C, depending on R, such that (4.71) 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 (2.9)-(2.10) by some constant multiplying (4.6) with, +for example, the same R. The relevant bound then follows from Proposition 4.1, and +that ǫ < 1. +Recall the notation introduced in (1.11)-(1.14), namely we can write +(y, x, ˆx1j) = (y, x2, . . . , xj−1, x, xj+1, . . . , xN). +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 (4.72) 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, +(4.73) +� +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. +(4.74) +Notice that when χR(|x − xj|) ̸= 0 we have |x − xj| < 2R. Take any θ ∈ (0, 1). Then +since φ ∈ C1,θ(R3N), we have local boundedness and local θ-H¨older continuity of ∇φ. +Therefore, by (2.9) and (2.10) there exists a constant C such that, when |x − xj| < 2R, +(4.75) +|∂m +y φ(y, x, ˆx1j)| ≤ ∥∇φ∥L∞(B((y,ˆx),2R)) ≤ Cf∞(y, ˆx; 4R) +(4.76) +��∂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; 4R). + +44 +PETER HEARNSHAW +The constant C depends on R and θ but is independent of x, y and ˆx. +Integration by parts in the variable xj is used in (4.73) to remove a single derivative +from ∂η +xj|x−xj|. Since ∂m +y φ(y, x, ˆx1j) has no dependence on xj, this process avoids taking +a second derivative of φ. Take any τ ≤ η with |τ| = 1. Integral (4.73) 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 (2.6), (4.50) and (4.75), can be bounded in absolute value by +C +� +R3N−3 λ(x, y, ˆx)−2f∞(x, ˆx; 4R)f∞(y, ˆx; 4R) +� +Φ(x, y, ˆx) + |∇Φ(x, y, ˆx)| +� +dˆx. +for some C depending on R and our choice of χ. The relevant bound then follows by +Corollary 4.3. +Using (2.6), (4.49), (4.50) and (4.76), the integral (4.74) can then be bounded in +absolute value by some constant multiplied by +� +R3N−3 λ(x, y, ˆx)−3+θf∞(x, ˆx; 4R)f∞(y, ˆx; 4R)Φ(x, y, ˆx) dˆx. +The relevant bound then follows by Lemma 4.2 with b = −3 + θ. +□ +Appendix A. 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. +We also set +(A.1) +g(x, y) = (x · y) ln +� +|x|2 + |y|2� +for x, y ∈ R3. For each x ∈ R3N we can then define the function +(A.2) +G(x) = K0 +� +1≤j 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. +Proof. To begin, we obtain bounds for derivatives of g(x, y) and G(x). It can be seen +that g ∈ C1,θ(R6) for all θ ∈ [0, 1). 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.5) +for all x, y ∈ R3. It follows that +(A.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, . . . , 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, . . . , yN) ∈ R3N with yk ̸= 0. Using the above inequality for every k ∈ P +and the definition of cluster derivatives, (1.15), we can obtain some C, depending on b, + +46 +PETER HEARNSHAW +such that +|Dη +P∇G(y)| ≤ CλP(y)−b +(A.7) +for all y ∈ Σc +P. Here, we also used that λP ≤ 1, the definition of λP in (1.21), and the +formula (1.20). Now, take some x ∈ Σc +P. As in Lemma 2.4, we use (1.22) to show that +(1 − r)λP(x) ≤ λP(y) for each y ∈ B(x, rλP(x)). Therefore, for C as in (A.7), we have +∥Dη +P∇G∥L∞(B(x,rλP (x))) ≤ C(1 − r)−bλP(x)−b +(A.8) +for all x ∈ Σc +P. +We now in a position to consider derivatives of φ = eGφ′. Firstly, we have ∇φ = +eGφ′∇G + eG∇φ′. 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′. +Taking the norm and using (A.6) and (A.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))) +� +. +(A.9) +for all x ∈ Σc +P. To the second term in the above bound we may then use Theorem +A.1 with constant C(r, R), followed by use of (A.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)) . +(A.10) +Together, (A.9) and (A.10) complete the proof. +□ +Acknowledgments. The author would like to thank A. V. Sobolev for helpful dis- +cussions in all matters of the current work. +References +[1] M. Reed and B. Simon. II: Fourier Analysis, Self-Adjointness. Elsevier, 1975. +[2] P. Hearnshaw and A.V. Sobolev. Analyticity of the one-particle density matrix. Ann. 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