diff --git "a/IdE2T4oBgHgl3EQfUQce/content/tmp_files/load_file.txt" "b/IdE2T4oBgHgl3EQfUQce/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/IdE2T4oBgHgl3EQfUQce/content/tmp_files/load_file.txt" @@ -0,0 +1,2098 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf,len=2097 +page_content='ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces Ian Marquette ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Junze Zhang †and Yao-Zhong Zhang ‡ School of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The University of Queensland Brisbane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' QLD 4072,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Australia January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2023 Abstract Superintegrable systems in 2D Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in the Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this paper, we obtain exact solutions via purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four different 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This is achieved by constructing the deformed oscillator realization and finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated by quadratic integrals respectively for each of the 12 superintegrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We also introduce generic cubic and quintic algebras, generated respectively by linear and quadratic integrals and linear and cubic integrals, and obtain their Casimir operators and deformed oscillator realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' As examples of applications, we present three classes of new superintegrable systems with cubic symmetry algebras in 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 1 Introduction Superintegrable systems of different orders have been attracting a large amount of international research activities, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [1], [2], [3], [4], [5], [6] , [7], [8] and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This paper is a contribution to the underlying algebraic structures and exact solutions of superintegrable systems in 2-dimensional (2D) curved spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Superintegrable systems in 2D spaces with constant or non-constant curvatures have been widely studied by means of separation of variables and St¨ackel transforms [10], [11], [12], [13], [14] and [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The St¨ackel transforms have been widely studied [16], [14], [17] provide useful tools in the classification of 2D superintegrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Through the method of the so-called coupling constant metamorphosis, St¨ackel transforms [18] enable one to establish the relationship between different superintegrable systems: they provide equivalence classes at the level of integrable and superintegrable Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' However, even if such Hamiltonians are connected via the St¨ackel transformations, they are distinct as Sturm-Liouville and spectral problem, and their exact solvability (with possibly different boundary conditions) and algebraic solutions need to be investigated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For a given superintegrable Hamiltonian which is separable in various coordinates, its solvability would in general depend on the coordinates used in the separation of variables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' it is exactly solvable in one coordinate system but only quasi-exactly solvable in another coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is well known that symmetry algebra structures play an important role in the analytic analysis of physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In the context of superintegrable models, the underlying symmetry algebra structures are usually polynomial algebras such as quadratic and cubic algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In [19], [2], [10], rank-1 quadratic algebra structures underlying certain 2D superintegrable systems, generated by integrals of motion of the ∗i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='marquette@uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='au †junze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='zhang@uqconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='au ‡yzz@maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='au 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='03810v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='SI] 10 Jan 2023 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES systems, were exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The authors in these references obtained the Casimir operator and deformed oscillator algebra realization of a generic quadratic algebra, and applied the relates to study the energy spectrum of the superintegrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In [14] and [11], examples of rank-1 cubic and quintic algebras in Darboux spaces were given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Higher order or higher rank polynomial algebras generated by integrals and their deformed oscillator algebra realizations were studied in [5], [20], [21], [22], [23], [24] , [25], [26], [27] and [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' More recently, by extending the Wigner-In¨on¨u method of Lie algebra contraction, the authors in [29], [30] showed that quadratic algebras from certain second-order superintegrable systems in 2D spaces are contractions of those with general 3-parameter potentials on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Superintegrable systems in 2D Darboux spaces were classified in [14][15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In 2 dimensions, there exist 4 possible Darboux spaces with metrics given by [31] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' d1(x, y) = (x + y) dxdy II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' d2(x, y) = � Ω (x − y)2 + Λ � dxdy III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' d3(x, y) = � Ω exp � −x + y 2 � + Λ exp(−x − y) � dxdy IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' d4(x, y) = Ω � exp � x−y 2 � + exp � y−x 2 �� + Λ exp � x−y 2 + exp � y−x 2 ��2 dxdy Here Ω, Λ ∈ R are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' According to the classification in [14][15], there exist 12 distinct classes of superintegrable systems with non-trivial potentials in the 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In each case, quadratic integrals of motion of the system were determined and were found to form a quadratic algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The wave functions and energy spectra of the systems were obtained by means of separation of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Superintegrable systems in Darboux spaces were also studied in [11] [32] [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It was shown there that free superintegrable systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' systems without potentials) in 2D and 3D flat conformal spaces are equivalent to systems in 2D and 3D Darboux spaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' However, as far as we know, finite dimensional representations of the polynomial algebras and algebraic derivations of the energy spectrum of the superintegrable systems have remained an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this paper we present a genuine algebraic approach to superintegrable systems in the 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The purpose of this paper is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' One is to give algebraic solutions to the existing 12 distinct classes of superintegrable systems in the four 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This is achieved by constructing the finite dimensional irreducible representation of the quadratic algebras underlying the 12 superintegrable systems via the deformed oscillator algebra techniques in [1] and [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' As one will see, energy spectrum for superintegrable systems in Darboux spaces are often determined by very complicated algebraic equations whose analytic and closed-form solutions can only be obtained by restricting the model parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The second purpose is to investigate superintegrable systems in 2D Darboux spaces with linear, quadratic or cubic integrals of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It was found in [11] that the free systems with linear and quadratic integrals in 2D Darboux spaces have cubic algebras as their underlying symmetry algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We will introduce generic cubic and quintic algebras, generated by linear and quadratic integrals and linear and cubic integrals, respectively, and construct their Casimir operators and deformed oscillator algebra realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We also present three classes of new superintegrable systems with non-trivial potentials in 2D Darboux spaces which have cubic algebras as their symmetry algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These superintegrable systems do not seem to belong to the families classified in [14][15] for systems with quadratic integrals in 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In Section 2, we obtain the Casimir operators, the deformed oscillator algebra realizations and finite-dimensional irreducible representations for the quadratic algebras generated by the quadratic integrals of motion of the 12 superintegrable systems in 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This enables us to give an algebraic derivation for the energy spectra of all the 12 classes of superintegrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In Section 3, we introduce generic cubic and quintic algebras generated by linear and higher order integrals of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We construct their Casimir operators and deformed oscillator algebra realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We also present examples of new superintegrable systems with linear and quadratic integrals in the 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In Section 4, we provide a summary of the main results of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES 2 Solutions of the 12 distinct classes of superintegrable systems in 2D Darboux spaces Consider a superintegrable system in a 2D Darboux space with coordinates (x, y) and metric gij(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The Hamiltonian of the system with potential V (x, y) is given by ˆH = 2 � j,k=1 1 � det(gjk) ∂ ∂xk �� det(gjk)gjk ∂ ∂xk � + V (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Let ˆX be an integral of motion (aka, constant of motion) of the system which commute with the Hamil- tonian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [ ˆX, ˆH] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' An integral of motion is said to be a polynomial in momenta of degree p, denoted by deg ˆX = p, if it has the form ˆX = p � j=0 rj(x, y) ∂p−j x ∂j y + s(x, y), where rj(x, y), s(x, y) are smooth functions in the coordinates x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In particular, integrals of motion of degree 1, 2 or 3 are usually called linear, quadratic or cubic integrals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Note that the Hamiltonian has degree 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' deg ˆH = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' As mentioned in the Introduction, superintegrable systems in the four 2D Darboux spaces with quadratic integrals of motion were classified in [14][15], and 12 distinct classes of potentials which pre- serve superintegrability were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this section, we present algebraic solutions to all the 12 existing superintegrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Note that in the following we will use the so-called separable coordinates in [14][15] for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' As indicated in [14][15], in such coordinates the parameters Ω, Λ in the metrics of the Darboux spaces can be conveniently absorbed into the model parameters of the systems by redefinition so that they do not appear explicitly in the expressions of Hamiltonians and integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 Darboux Space I According to the classification in [14][15], in the Darboux space I, there are two possible superintegrable systems with potentials given by V1(x, y) = b1(4x2 + y2) 4x + b2 x + b3 xy2 , V2(x, y) = a1 x + a2y x + a3(x2 + y2) x , respectively, where bi, ai are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 Potential V1(x, y) For superintegrable system in Darboux space I with the Hamiltonian ˆH = 1 4x � ∂2 x + ∂2 y � + V1(x, y) asso- ciated to V1, the constants of motion are given by [15] A = ∂2 y + 4b3 y2 + b1y2, B = y∂y∂x − x∂2 y + ∂x 2 − y2 4x � ∂2 x + ∂2 y � + b1y4 4x + b2y2 x + b3(4x2 + y2) y2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These integrals satisfy the following quadratic algebra relations [A, B] = C, [A, C] = −8 ˆHA − 16b1B, [B, C] = 6A2 + 8 ˆHB + 16b2A − 2b1(3 + 16b3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES This is the symmetry algebra of the superintegrable system The Casimir of this algebra is given by K1 = C2 − 4A3 + 8 ˆH{A, B} − 16b2A2 − 16b1B2 + 4b1(11 + 16b3)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We can show that with the differential realization of A, B the Casimir K1 has the following form in terms of the Hamiltonian ˆH, K1 = −4(3 + 16b3) ˆH2 + 16b1b2(3 + 16b3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In order to obtain the energy spectrum of the system via algebraic means, we now construct realization of the quadratic algebra in terms of the deformed oscillator algebra of the form [N, b†] = b†, [N, b] = −b, bb† = Φ(N + 1), b†b = Φ(N), (1) where N is the number operator and Φ(z) is a well-defined real function satisfying Φ(0) = 0, Φ(z) > 0, ∀z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (2) Φ(x) is called the structure function of the deformed oscillator algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is non-trivial to obtain such a realization and the corresponding structure function Φ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After a long computation, we find in the present case that A = 4 � −b1 (N + η), B = 2 ˆH √−b1 (N + η) + b† + b map the quadratic algebra to the deformed oscillator algebra with structure function given by Φ(I) 1 (N, η) = − 1 16b1 � −4(N + η)16b1b2 + (−b1)3/2(16b3 + 11) − 4 ˆH2 + 2b3/2 1 (16b3 + 3) +64 � −b1b1(N + η)3 + 16(N + η)2(4b1b2 − ˆH2) − (16b3 + 3)(2b1 � −b1 − 4b1b2 + ˆH2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Here η is a constant to be determined from the constraints on the structure function Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We now obtain the finite-dimensional unitary irreducible representations (unirreps) of the deformed oscillator algebra in the Fock space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Let |z, E⟩, denote the Fock basis states labelled by the eigenvalues z and E of N and ˆH, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Acting the structure function on the Fock states, we find that it is factorized to the following form ΦI 1(z, η) = � z + η − 1 4 � 2 − � 1 − 16b3 �� � z + η − 1 4 � 2 + � 1 − 16b3 �� � z + η + 2b1 �√−b1 − 2b2 � + E2 4(−b1)3/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For the unirreps to be finite dimensional, we impose the following constraints on the structure function, Φ(0, η) = 0, Φ(p + 1, η) = 0, (3) where p is a positive integer, p = 0, 1, 2, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These constraints give (p+1)-dimensional unirreps in the Fock space and their solutions give the constant η and energy spectrum E of the underlying superintegrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' There are two sets of solutions from the constraints on the structure function: η = 1 4 � 2 + ϵ√1 − 16a2 � , Eim = ±2 √ −1 (−b1)3/4 � p + 1 − ϵ 4 � 1 − 16b3 + b2 √−b1 , 4 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES and η = −2b1 �√−b1 − 2b2 � + E2 4(−b1)3/2 , Eϵ = ±2(−b1)3/4 � p + 1 + ϵ 4 � 1 − 16b3 − b2 √−b1 , where ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The first set of solutions give complex energies which are not physical and thus will be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' So the energy spectrum of the system is given by the second set of solutions which are real for ϵ = +1, b1 < 0, b2 ≤ 0, b3 < 1/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure function for the corresponding (p + 1)-dimensional unirreps is Φ(I) E+(z) = z(z − p − 1) � z − 2b1 �√−b1 − 2b2 � + E2 + 4(−b1)3/2 − 1 4 � 2 + � 1 − 16b3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In the following subsections, we would only give the values of parameter η which can lead to real energies E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 Potential V2(x, y) Constants of motion for the superintegrable system in Darboux space I with the Hamiltonian ˆH = 1 4x � ∂2 x + ∂2 y � + V2(x, y) corresponding to the potential V2 are given by [15] A = y∂y∂x − x∂2 y + ∂x 2 − y2 4x � ∂2 x + ∂2 y � − 2a2y x + 2a2(x2 − y2) x + 2a2y(x2 − y2) x , B = ∂2 y + 4a2y + 4a3y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' They satisfy the following quadratic algebra relations [A, B] = C, [A, C] = 16a2 ˆH − 16a3B, [B, C] = 16a3A + 8(a2 2 + 4a1a3) − 8 ˆH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The Casimir operator of the algebra is given by K2 = C2 + 16a3A2 + 16a3B2 − 32a2 ˆHB + 16 � (a2 2 + 4a1a3) − ˆH2� A, which in terms of the differential realization of A, B takes the constant value K2 = 64(a2 3 − a1a2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We then determine the realization of above quadratic algebra in terms of the deformed oscillator algebra (1) and apply its finite dimensional unirreps to obtain the energy spectrum of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After computations, we find that A = 4√−a3(N + η), B = a2 ˆH a3 + b† + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' transform the quadratic algebra to the deformed oscillator algebra with structure function Φ(I) 2 (N, η) = � 4a1a3 + a2 2 − ˆH2�2 16a2 3 − a1a2 2 a3 − 1 12(N + η) � 24a2 ˆH √−a3 + 48a3 � + a2 ˆH √−a3 + 4a3(N + η)2 + a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Moreover, the action of this structure function on the Fock states |z, E⟩ is factorized as Φ(I) 2 (z, η) = � z + η − m+(E) + 2a3 4a3 � � z + η − m−(E) + 2a3 4a3 � , 5 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES where η is a constant to be determined and m±(E) = a2E √−a3 ± � 64a2 1a2 3 + 4a1 �a2 2(8a3 − 1) − 8a3E2� + 4a4 2 − a2 2(8a3 + 1)E2 a3 + 4E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We now impose the constraints (3) to obtain finite-dimensional unirreps of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We find that for p = 0, 1, 2, · · · , we have Case 1: η−(E) = 1 4a3 (m−(E) + 2a3) and � 64a2 1a2 3 + 4a1 �a2 2(8a3 − 1) − 8a3E2� + 4a4 2 − a2 2(8a3 + 1)E2 a3 + 4E4 = 2a3(p + 1), (4) which has solutions only for a3 > 0 and the energy spectrum of the system is given by E+a3 = ± 1 √8a3 �� 128a1a2 2a2 3 + a4 2(16a3 + 1) + 64a4 3(p + 1)2 + 32a1a2 3 + a2 2(8a3 + 1), Notice that E+a3 is real for a1 > 0, a3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 2 η+(E) = 1 4a3 (m+(E) + 2a3) and � 64a2 1a2 3 + 4a1 �a2 2(8a3 − 1) − 8a3E2� + 4a4 2 − a2 2(8a3 + 1)E2 a3 + 4E4 = −2a3(p + 1), which has solutions only for a3 < 0 and the energies of the system are E−a3 = ± 1 √−8a3 �� 128a1a2 2a2 3 + a4 2(16a3 + 1) + 64a4 3(p + 1)2 − 32a1a2 3 − a2 2(8a3 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (5) Obviously for a3 < 0 there exist ranges of model parameters a1, a2 such that the eneries E−a3 of the system are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure function for both cases 1 and 2 corresponding to the (p + 1)-dimensional unirreps of the algebra is given by Φ(I) E±a3(z) = z(z − p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 Darboux Space II In the Darboux space II, there are three superintegrable systems with potentials given by [14] V1(x, y) = x2 x2 + 1 � a1 � x2 4 + y2 � + a2y + a3 x2 � , V2(x, y) = x2 x2 + 1 � b1(x2 + y2) + b2 x2 + b3 y2 � V3(x, y) = c1 + c2 x2 + c3 y2 x2 + y2 + 1 x2 + 1 y2 , respectively, where aj, bj, cj are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 Potential V1(x, y) The constants of motion of the superintegrable system in Darboux space II with the Hamiltonian ˆH = x2 x2+1 � ∂2 x + ∂2 y � + V1(x, y) associated to the potential V1 are A = ∂2 y + a1y2 + a2y, B = 2y x2 + 1 � ∂2 y − x2∂2 x � + 2x∂x∂y + ∂y + a1 2 y � x2 + x2 + 4y2 x2 + 1 � + a2 2 � x2 + 4y2 x2 + 1 � − 2a3y x2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 6 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES They satisfy the following quadratic algebra relations [14] [A, B] = C, [A, C] = −4a1B − 4a2A, [B, C] = −24A2 + 4a2B + 32 ˆHA − 8 ˆH2 − 8a1 ˆH + 6a1 + 8a1a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Its Casimir operator can be shown to be given by K1 = C2 − 16A3 + 4a1B2 + 4a2{A, B} + � 4a1(4a3 − 11) − (16a1 ˆH + 16 ˆH2) � A + 32 ˆHA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In term of the differential realization of A, B, the Casimir K1 takes the simple form K1 = (32a1+4a2 2) ˆH− a2 2(3 + 4a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By computations similar to those in the previous subsection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we find that A = 2√−a1(N + η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = 2a2 √−a1 (N + η) + a2 ˆH a1 + b† + b map the quadratic algebra to the deformed oscillator algebra (1) with structure function given by Φ(II) 1 (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = − 12a3 1 − 3√−a1a1a2 2 − 16a3 1a3 − 4√−a1a1a2 2a3 + 32√−a1a2 1 ˆH + 16a3 1 ˆH − 8a1a2 2 ˆH + 4√−a1a1a2 2 ˆH + 16a2 1H2 + 4√−a1a2 2H2 + (N + η) � 88a3 1 + 16√−a1a1a2 2 + 32a3 1a3 − 128√−a1a2 1 ˆH − 32a3 1 ˆH + 16a1a2 2 ˆH − 32a2 1 ˆH2� + (N + η)2 � −192a3 1 − 16√−a1a1a2 2 + 128√−a1a2 1 ˆH � + 128a3 1(N + η)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Here η is a constant to be determined from the constraints of the structure function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Acting on the Fock basis states |z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' E⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' the structure function Φ(II) 1 becomes factorized Φ(II) 1 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = � z + η − f1(E) + ω(E) + f2(E) 24a3 1 � � z + η − 1 96a3 1 � 4f1(E) − 2 � 1 − i √ 3 � ω(E) + � 1 + i √ 3 � f2(E) �� � z + η − 1 96a3 1 � 4f1(E) − 2 � 1 + i √ 3 � ω(E) + � 1 − i √ 3 � f2(E) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where f1(E) =a1 � 12a2 1 + √−a1a2 2 + 8(−a1)3/2E � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' f2(E) = 1 ω(E) � a3 1(12a3 1(4E − 4a3 + 1) − a4 2 − 16a2 1E2 − 8a1a2 2E) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ω(E) = 3� τ1(E) + τ2(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' τ1(E) =6a6 1 � a4 2 + 8a1Ea2 2 + 16a2 1E2 + a3 1(−16a3 + 16E + 4) � � 3(4a3 − 4E − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' τ2(E) =a1a6 2(−a1)7/2 + 12a4 2E(−a1)11/2 − 48a2 2E2(−a1)13/2 − 4 � 9a2 2(4a3 − 4E − 1) − 16E3� (−a1)15/2 − 144E(−4a3 + 4E + 1)(−a1)17/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' To determine the constant η and energy spectrum E of the superintegrable system, we impose the constraints (3) which give (p + 1)-dimensional unirreps of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We find Case 1: The constant η is given by η1(E) = f1(E) + ω(E) + f2(E) 24a3 1 7 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES and the energy E satisfies the algebraic equation, ω(E) + f2(E) + 1 2 � 1 − ϵ i √ 3 � ω(E) − 1 4 � 1 + ϵ i √ 3 � f2(E) = −24(p + 1)a3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (6) Case 2: η2(E) = 1 96a3 1 � 4f1(E) − 2 � 1 − ϵ i √ 3 � ω(E) + � 1 + ϵ i √ 3 � f2(E) � and the energy is determined by ω(E) + f2(E) + 1 2 � 1 − ϵ i √ 3 � ω(E) − 1 4 � 1 + ϵ √ 3i � f2(E) = 24(p + 1)a3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (7) In both cases above, ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The energy spectrum E of the system are obtained by solving the algebraic equations (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' However, it is in general very difficult to obtain analytical solutions of these equations, due to their complicated form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' To demonstrate that these equations have real solutions, we have a closer look at restricted model parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Without the loss of generality, we consider the case where −a1 = a2 = a3 = a for any a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For such model parameters, the structure function has the simple form, Φ(II) 1 (z, η) = � z + η − 1 8 �√a + 4 �� � z + η − 1 4a � 2√aE + 2a − a √ 4E + 1 − 4a �� � z + η − 1 4a � 2√aE + 2a + a √ 4E + 1 − 4a �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3) on the structure function lead to the determination of constant η and energy E of the superintegrable system for the model parameters −a1 = a2 = a3 = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' There are two sets of solutions: One is that η = 1 8 (√a + 4) and Eϵ = 1 4 � 8√a(p + 1) + 3a + 2ϵ � 8a3/2(p + 1) − 2a2 + a � , where ϵ = ±1, with the associated structure function Φ(II) Eϵ (z) = z(p + 1 − z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The energy spectrum Eϵ is real for 0 < a ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The second set of solutions is given by η(E) = 1 4a � 2√aE + 2a − a √ 4E + 1 − 4a � and the corresponding energy spectrum of the system and structure function for the (p + 1)-dimensional unirreps of the deformed oscillator algebra are given by E = p(p + 2) + a + 3 4, Φ(II) E (z) = z(z − p − 1) � z + 1 8a � 3a3/2 − 4a(p + 1) + √a(4p2 + 8p + 3) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Thus we have demonstrated that there exist indeed non-trivial model parameters which give real energies of the superintegrable system in both Case 1 and Case 2 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 Potential V2(x, y) The superintegrable system in Darboux II with potential V2(x, y) has Hamiltonian ˆH = x2 x2+1 � ∂2 x + ∂2 y � + V2(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This system possesses the following integrals of motion [14] A = ∂2 y + b1y2 + b3 y2 , B = (y2 − x4)∂2 y + x2(1 − y2)∂2 x x2 + 1 + 2xy∂x∂y + x∂x + y∂y − 1 4 + x2 + y2 x2 + 1 � b1(x2 + y2) − b2 − b3 x2 y2 � , 8 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES which form the quadratic algebra relations [A, B] = C, [A, C] = 8A2 − 16b1B + 16b1 ˆH − 16b1(b2 + b3 + 3 4), [B, C] = −8{A, B} + 8 ˆHB + 12A − 8 ˆH2 + 8(b2 − b3 − 3 4) ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By a direct calculation, we find the Casimir operator of the algebra K2 =C2 − 8{A2, B} + 8 ˆH{A, B} + 16b1B2 + 76A2 + � 16(b3 − b2 + 19 4 ) ˆH − 16 ˆH2 � A + � 8b1(4(b2 + b3) + 3) − 32b1 ˆH � B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This Casimir operator can be expressed in terms of Hamiltonian as K2 = −16 � b1 + b3 + 3 4 � ˆH2 − 8b1(4b3 − 4b2 + 3) ˆH + b1 � 36 + 48b3 − (4b3 − 4b2 + 3)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It can be shown that after the change of basis A = 4 � −b1(N + η), B = 8(N + η)2 − 2 ˆH √−b1 − 16(b2 + b3 + 3 4)(N + η) − b1 ˆH b1 + b† + b, the quadratic algebra becomes the deformed oscillator algebra with structure function Φ(II) 2 (N, η) = 1 16 � 4b3 + 16N 2 + 16N(2η − 1) + 16η2 − 16η + 3 � � 4b2 + 1 − 4 ˆH �√−b1 + 2N + 2η − 1 � √−b1 + 16N 2 + 32Nη − 16N + 16η2 − 16η + 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' On the Fock states |z, E⟩, the structure function is factorized as follows Φ(II) 2 (z, η) = � z + η − 1 4 � 2 − � 1 − 4b3 �� � z + η − 1 4 � 2 + � 1 − 4b3 �� � z + η − 2b1 − γ+(E) 4b1 � � z + η − 2b1 − γ−(E) 4b1 � , where γ±(E) = � b2 1(4E − 4b2 + 1) ± � −b1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3), for any p ∈ N+ we get the following values for the parameter η and energy E: Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η+(E) = 1 4b1 (2b1 − γ+(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This η value gives the following energy spectrum of the system and the corresponding structure function of the deformed oscillator algebra E− = −(p + 2) � −b1, Φ(II) E− (z) = z(z − p − 1) � z + 1 4b1 � b1 � 1 − 4b3 − γ+(E−) �� � z − 1 4b1 � b1 � 1 − 4b3 + γ+(E−) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The enegry E− is real for b1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η−(E) = 1 4b1 (2b1 − γ−(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This η value gives two sets of energies of the system, E+ =(p + 2) � −b1, (8) Eϵ = − 2b1 + 4 � −b1 � p + 1 + ϵ 4 � 1 − 4b3 � ± � 4b2 1 − 16b1 � −b1 � p + 1 + ϵ 4 � 1 − 4b3 � + 4b1b2 − b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (9) 9 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES Eϵ above is obtained by solving the algebraic equation γ−(E) = 4b1 �p + 1 + ϵ 4 √1 − 4b3 � from the con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (Notice that the other algebraic equation γ+(E) = 4b1 �p + 1 + ϵ 4 √1 − 4b3 � lead to complex solutions and its solutions are not shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=') Obviously E+ is real for b1 < 0 and Eϵ is real for ϵ = +1, b1 < 0, b2 < 1 4, b3 < 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure functions corresponding to E+, Eϵ in the Case 2 above are given by Φ(II) E+ (z) = z(z − p − 1) � z + 1 4b1 � b1 � 1 − 4b3 − γ−(E+) �� � z − 1 4b1 � b1 � 1 − 4b3 + γ−(E+) �� , Φ(II) Eϵ (z) = z(z − p − 1) � z + 1 2b1 � −b1 Eϵ � � z − 1 4b1 � γ−(Eϵ) + ϵb1 � 1 − 4b3 �� , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 3: η = 1 4 �2 + ϵ√1 − 4b3 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The corresponding energies are given the same expression as Eϵ above (and are obtained from solving the algebraic equation γ+(E) = −4b1 �p + 1 + ϵ 4 √1 − 4b3 �).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 Potential V3(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' y) The constants of motion for the superintegrable system in Darboux space II with the Hamiltonian ˆH = x2 x2+1 � ∂2 x + ∂2 y � + V3(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' y) associated to the potential V3 are given by A = � y2 + 1 y2 � ∂2 x − � x2 + 1 x2 � ∂2 y x2 + y2 + 1 x2 + 1 y2 + c1x2(y4 + 1) + c2(y4 + 1) − c3(x4 + 1) (x2y2 + 1)(x2 + y2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B =c1(x2 + y2) − c2(y4 − 1) − c3(x4 − 1) 4(x2y2 + 1) + xy(x2 − y2) � xy∂2 x − xy∂2 y + (x2 − y2)∂x∂y � + 1 x2y2 + 1 �� x2 − y2 4 + y4 � x2∂2 x + � x2 − y2 4 + x4 � y2∂2 y + 2xy � x2 − y2 2 − x2y2 � ∂x∂y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' They form the following quadratic algebra relations [A, B] = C, [A, C] = 2A2 + 2c1A + 16 ˆHB + 6 ˆH − 8 ˆH2, [B, C] = −2{A, B} + (c2 + c3)A − c1c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The Casimir operator of this algebra is K3 = C2 − 2{A2, B} − 16 ˆHB2 + (c2 + c3 + 4)A2 + 2c1{A, B}a − 2c1(c3 + 2)A + (16 ˆH2 − 12 ˆH)B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' With the differential realization of A, B, the Casimir operator can be expressed in terms of the Hamilto- nian as K3 = 4(c2 + c3) ˆH2 + (c2 1 − 4c2c3 − 3(c2 + c3)) ˆH − 3 + 4c3 4 c2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We can convert the quadratic algebra into the deformed oscillator algebra by using the realization A = 4 � ˆH(N + η), B = −2(N + η)2 + c1 2 � ˆH (N + η) − 3 ˆH − 4 ˆH2 8 ˆH + b† + b with the corresponding structure function given by Φ(II) 3 (N, η) = − 1 256 ˆH � 4c3 − 4 ˆH + 16N 2 + 32Nη − 16N + 16η2 − 16η + 3 � × � −c2 1 + 4c1 � ˆH(2N + 2η − 1) + ˆH � −4c2 + 4 ˆH − 16N 2 − 32Nη + 16N − 16η2 + 16η − 3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 10 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES By acting Φ(II) 3 on the Fock basis states |z, E⟩, we find that the structure function is factorised as Φ(II) 3 (z, η) = � z + η − 1 4 � 2 − � −4c3 + 4E + 1 �� � z + η − 1 4 � 2 + � −4c3 + 4E + 1 �� × � z + η − 1 4E � −E � −4c2 + 4E + 1 + c1 √ E + 2E �� × � z + η − 1 4E � E � −4c2 + 4E + 1 + c1 √ E + 2E �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We now obtain the energy spectrum of the system from the finite-dimensional unirreps of the deformed oscillator algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3) which give (p + 1)-dimensional unirreps for any p ∈ N+, we determine the parameter η and the energy E of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' There are two sets of solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 1: η(E) = 1 4 �2 − √−4c3 + 4E + 1 � and the energies are determined by either � −4c3 + 4E + 1 − 2(p + 1) = 0 (10) or c1 √ E + � −4c3 + 4E + 1 + � −4c2 + 4E + 1 = 4(p + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (11) Solution to the algebraic equation (10) gives the energies Ec3 = p(p + 2) + c3 + 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure function of the corresponding (p + 1)-dimensional unirreps is Φ(II) Ec3 (z) =z(z − p − 1) � � � �z − 1 2 � p + 1 − � (p + 1)2 + c3 − c2 � − c1 4 �� p + 1 2 � � p + 3 2 � + c3 � � � � � � � �z − 1 2 � p + 1 + � (p + 1)2 + c3 − c2 � − c1 4 �� p + 1 2 � � p + 3 2 � + c3 � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Other possible energies of the system are given by solutions to the algebraic equation (11), which read E± = 1 4 (p + 1 + c1)2 � 2c2 + 2c3 − 1 ± � (p + 1 + c1)2 + 4c2c3 − (c2 + c3) + 1 4 � (p + 1 + c1)2 − (c2 − c3)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These energies are real for the model parameters satisfying 4c2c3 + 1 4 > c2 + c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The corresponding structure functions for the (p+1)-dimensional unirreps of the algebra are Φ(II) E± (z) =z(z − p − 1) � z − 1 2 � −4c3 + 4E± + 1 � � z − 1 4 �� −4c3 + 4E± + 1 − � −4c2 + 4E± + 1 + c1 √E± �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 2: η(E) = 1 4E � c1 √ E + 2E − E√−4c2 + 4E + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This η value gives the following energy spectrum of the system and the corresponding structure function of the unirreps, Ec2 = p(p + 2) + c2 + 3 4, Φ(II) Ec2 (z) =z(z − p − 1) � � � �z + 1 2 � p + 1 − � (p + 1)2 + c2 − c3 � + c1 4 �� p + 1 2 � � p + 3 2 � + c2 � � � � � � � �z − 1 2 � p + 1 + � (p + 1)2 + c2 − c3 � + c1 4 �� p + 1 2 � � p + 3 2 � + c2 � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 11 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 Darboux Space III In Darboux space III, there exist 4 different potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In terms of the separable coordinates (u, v) and (µ, ν), they are given by V1(u, v) = a1u + a2v + a3 4 + u2 + v2 , V2(u, v) = b1 u2 + b2 v2 + b3 4 + u2 + v2 , V3(µ, ν) = c1(µ + ν) + c2 µ+ν µν + c3 ν2−µ2 ν2µ2 (µ + ν)(2 + µ − ν) , V4(µ, ν) = d1µ + d2ν + d3ν2 + µ2 (µ + ν)(2 + µ − ν) , where ai, bi, ci, di are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 Potential V1(u, v) The constants of motion of the superintegrable system in Darboux space III with the Hamiltonian ˆH = exp(2u) 4(exp(u))+1 �∂2 u + ∂2 v � + V1(u, v) associated to the potential V2 are given by [14] A = (2 + v2)∂2 u − (2 + u2)∂2 v 2(4 + u2 + v2) + a1u(2 + v2) − 2a2v(2 + u2) + a3(v2 − u2) 4(4 + u2 + v2) , B = 2uv (∂2 u + ∂2 v) 2(4 + u2 + v2) − 2∂u∂v + a1v(v2 − u2 + 4) + a2u(u2 − v2 + 4) − 2a3vu 4(4 + u2 + v2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' They form the quadratic algebra with the commutation relations [A, B] = C, [A, C] = ˆHB − a2a1 8 , [B, C] = − ˆHA − a2 2 − a2 1 16 , which is the symmetry algebra of the superintegrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By a direct computation, we obtain the Casimir operator of this algebra K1 = − ˆHA2 − ˆHB2 − a2 2 − a2 1 8 A + a1a2 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We can show that in terms of the Hamiltonian this Casimir operator takes the form K1 = − ˆH3 + 1 2(a3 + 1 2) ˆH2 + 1 16(2a2 1 + 2a2 2 − a2 3) ˆH − a3(a2 1 + a2 2) 32 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' To determine the energy spectrum of the system, we now construct the deformed oscillator algebra realization of the quadratic algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We find that A = � ˆH(N + η), B = a1a2 8 ˆH + b† + b, trsansform the quadratic algebra into the deformed oscillator algebra with the structure function Φ(III) 1 (N, η) = 1 256 ˆH � a2 1 + 2a3 ˆH − 4 ˆH3/2 � 2 � ˆH + 2N + 2η − 1 �� × � a2 2 + 2a3 ˆH + 4 ˆH3/2 � −2 � ˆH + 2N + 2η − 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Here η is a constant parameter to be determined from the constraints (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Acting on the Fock basis states |z, E⟩, the structure function Φ(III) 1 becomes Φ(III) 1 (z, η) = � z + η − 1 8E3/2 � a2 1 + 2E � a3 − 4E + 2 √ E ��� × � z + η + 1 8E3/2 � a2 2 + 2E(a3 − 4E − 2 √ E) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 12 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES The constraints (3) give the (p + 1)-dimensional unirreps of the deformed oscillaor algebra and their solutions determine the constant η and energy spectrum of the superintegrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' There are two sets of solutions: Case 1: η(E) = 1 8E3/2 � a2 1 + 2E � a3 − 4E + 2 √ E �� and energies E are determined by the algebraic equation 8E3/2 (p + 1) + 4a3E + a2 1 + a2 2 = 16E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (12) Case 2: η(E) = − 1 8E3/2 � a2 2 + 2E � a3 − 4E − 2 √ E) �� and energies E satisfy 16E2 + 8E3/2 (p + 1) = 4a3E + a2 1 + a2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (13) The algebraic equations (12) and (13) can be solved by using symbolic computation packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It can be shown that there exist model parameters ai such that solutions to these algebraic equations for energies are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' To demonstrate this, we consider the case in which the model parameters satisfy a1 = 0 and a2 = a3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this case we find that the structure function reduces to Φ(III) 1 (z, η) = � z + η − �1 2 − √ E + 1 4 √ E �� � z + η − �1 2 + 1 8E3/2 (8E2 − 2E − 1) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3) gives the constant η and energies as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η(E) = 1 2 − √ E + 1 4 √ E which leads to the algebraic equation 4E(1 − 4E) + 1 + 8E3/2(p + 1) = 0 for E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This equation has real solution given by E± = 1 48 � 3p2 + √ 3 g(p) + 6p + 9 ± � 6 f(p) � , where e(p) = 3 � 27p4 + 108p3 + 252p2 + 3 √ 3 � (p + 1)4 (27p4 + 108p3 + 310p2 + 404p + 575) + 288p + 367 g(p) = � 3 (p2 + 2p + 3)2 + 2 × 22/3e(p) + 1 e(p) 4 3√ 2 (6p2 + 12p + 31) + 8 f(p) = 3 � p2 + 2p + 3 �2 − 22/3e(p) − 1 e(p)2 3√ 2 � 6p2 + 12p + 31 � + 1 g(p) 3 √ 3(p + 1)2 � p4 + 4p3 + 12p2 + 16p + 23 � + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is clear that g(p) is real for all p ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We now show that f(p) > 0 for all p ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Let f0(p) = 3 � p2 + 2p + 3 �2 − 22/3e(p) − 1 e(p) 2 3√ 2 � 6p2 + 12p + 31 � + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By using symbolic computation package, we found that df0(p) dp > 0 for all p ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Hence f0(p) is strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Moreover, f0(0) = −62 3� 2 15 √ 69+367 − 22/3 3� 15 √ 69 + 367 + 35 ∼= 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='5741 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It follows that f(p) > 0 for all p ∈ N+ and the energy E given above is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η(E) = 1 2 + 1 8E3/2 (8E2−2E−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This leads to the algebraic equation 4E(4E−1)−1+8E3/2(p+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It gives the same energy expression as in Case a above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For both case a and case b above, the structure function corresponding to the (p + 1)-dimensional unirreps of the deformed oscillator algebra is simply Φ(III a,b (z) = z(z − p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 13 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 Potential V2(u, v) The integrals of motion of the superintegrable system associated to the potential V2 with Hamiltonian ˆH = exp(2u) 4(exp(u))+1 �∂2 u + ∂2 v � + V2(u, v) in Darboux space III are given by [14], A = u2∂2 v − 2uv∂u∂v + v2∂2 u + b1v2 4u2 + b2u2 4v2 , B = (2 + v2)∂2 u − (2 + u2)∂2 v 2(4 + u2 + v2) + 2b1v2(v2 + 2) − 2b2u2(u2 + 2) + b3(v2 − u2) 4(4 + u2 + v2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These integrals form the quadratic algebra of the form [A, B] = C, [A, C] = −2{A, B} − (b1 + b2 + 1)B + (b1 − b2) ˆH + (b2 − b1)b3 4 , [B, C] = −2B2 − (b1 + b2 + 1)B + (b1 − b2) ˆH + (b2 − b1)b3 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By a direct calculation, we find the Casimir operator of the algebra K2 = C2 + 2{A, B2} + (b1 + b2 + 5)B2 − 4 ˆHA2 − 2(b1 − b2) ˆHB − b3(b2 − b1)B − 4 ˆHA + (2b3 − 1) ˆHA − b2 3 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' With the differential realization of A, B and in terms of ˆH, the Casimir K2 takes the simple form K2 = −(b1 + b2 − 2) ˆH2 + � (b3 + 3 2)(b1 + b2) 2 − b3 − b1b2 − 1 2 � ˆH − b2 3(b1 + b2 − 2) 16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The quadratic algebra can be transformed into the deformed oscillator algebra via the realization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=', change of basis) A = − � (N + η)2 − 1 4 + b1 + b2 + 1 4 � , B = −(b1 − b2) ˆH + (b2−b1)b3 4 16 � (N + η)2 − 1 4 � + b†ρ(N) + ρ(N)b, where ρ(N) = 1 3 · 212 · (−2)8(N + η)(1 + N + η)(1 + 2(N + η))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure function is given by Φ(III) 2 (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = 4096((2N + 2η − 1)2 � b2 2(3b1 + 3b2 + 7) − 4 ˆH � 3b2 1 + b1(12b2 − 12 ˆH + 11) +9b2 2 − 12b2 ˆH + 25b2 − 28 ˆH + 4 �� − 48(1 − 2(N + η))2 � − 1 16b2 2(b1 + b2 − 2) +1 2 ˆH �� b2 + 3 2 � (b1 + b2) − 2b1b2 − 2b2 − 1 � − ˆH2(b1 + b2 − 2) � + (2N + 2η − 1)2 � 12N 2 + 12N(2η − 1) + 12η2 − 12η − 1 � × � b2 2 − 4 ˆH(2b1 + 4b2 − 4 ˆH + 1) � + 12(b1 − b2)2(b2 − 2 ˆH)2 − 12 ˆH(2(N + η) − 3)(2(N + η) + 1)(1 − 2(N + η))4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Here η is a constant to be determined from the constraints on the structure function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 14 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES By acting on Fock basis states |z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' E⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we can show that the structure function Φ(III) 2 is factorized as Φ(III) 2 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='6 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='δ1(E) + (b3 − 4E)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 8(b1 + b2) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='g(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='Eδ1(E) + 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3)g(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='Eδ1(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where f(E) = 2 � b2 3 − 8Eb3 − 8(b1 + b2 − 2E)E � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' g(E) = b4 3 − 16Eb3 3 + 4E(2b1 + 2b2 + 24E + 3)b2 3 − 32E2(2b1 + 2b2 + 8E + 3)b3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' + 16E2(b2 1 + 14b2b1 + 8(E − 3)b1 + b2 2 + 16E2 + 8b2(E − 3) + 12E + 15),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' δ1(E) = 3� ρ1(E) + ρ2(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ρ1(E) = b6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 24Eb5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 240E2b4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 12b1Eb4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 12b2Eb4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 18Eb4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 1280E3b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 192b1E2b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 192b2E2b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 288E2b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 3840E4b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 1152b1E3b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 1152b2E3b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 1728E3b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 48b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1E2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 48b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2E2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 288b1E2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 480b1b2E2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 288b2E2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 360E2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 6144E5b3 − 3072b1E4b3 − 3072b2E4b3 − 4608E4b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 384b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1E3b3 − 384b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2E3b3 + 2304b1E3b3 + 3840b1b2E3b3 + 2304b2E3b3 − 2880E3b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 4096E6 + 3072b1E5 + 3072b2E5 + 4608E5 + 768b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1E4 + 768b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2E4 − 4608b1E4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 7680b1b2E4 − 4608b2E4 + 5760E4 + 64b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1E3 + 64b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2E3 + 3744b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1E3 − 2112b1b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2E3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 3744b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2E3 − 8064b1E3 − 2112b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b2E3 + 10944b1b2E3 − 8064b2E3 + 3456E3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ρ2(E) = 128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2043 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 8Eb3 − 8(b1 + b2 − 2E)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�2 − 12E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(2b1 + 2b2 + 1)b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−8(2b1 + 2b2 + 1)Eb3 − 4E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 2(b2 + 4E − 4)b1 + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + 8b2 − 8b2E − 4E − 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='���3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 262144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 24Eb5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 6E(2b1 + 2b2 + 40E + 3)b4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 32E2(6b1 + 6b2 + 40E + 9)b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 24E2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 4(5b2 − 12E + 3)b1 + 2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + 160E2 + 72E + 12b2(4E − 1) + 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 192E3 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 4(5b2 − 4E + 3)b1 + 2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + 32E2 + 24E + 4b2(4E − 3) + 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 32E3 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 + (−66b2 + 24E + 117)b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='11b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + (40E − 57)b2 − 16E2 + 24E + 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+2b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + 3b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2(8E + 39) + 12b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='8E2 − 12E − 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='32E3 + 36E2 + 45E + 27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='���2� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3) which give the (p + 1)-dimensional unirreps of thedeformed oscillator algebra, we determine the constant η and obtain the following algebraic equations for the energies E: 15 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η1(E) = 1 12 � 6 − √ 3 � δ1(E)+(b3−4E)2 E − 8(b1 + b2) + g(E) Eδ1(E) + 12 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This η value gives five sets of algebraic equations, δ1(E) + (b3 − 4E)2 + g(E) δ1(E) = (12p(p + 2) + 8(b1 + b2)) E, η1(E) − 1 24 � �12 + ϵ √ 6 � f(E) E + (−1 + i √ 3)δ1(E) E − (1 + i √ 3)g(E) Eδ1(E) + 24 � � + p + 1 = 0, η1(E) − 1 24 � �12 + ϵ √ 6 � f(E) E − (1 + i √ 3)δ1(E) E + (−1 + i √ 3)g(E) Eδ1(E) + 24 � � + p + 1 = 0, where ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Real solutions to each algebraic equation above give the energies of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η2(E) = 1 24 � 12 − √ 6 � f(E) E + i(i+ √ 3)δ1(E) E − i(−i+ √ 3)g(E) Eδ1 + 24 � and energy spectra from the real solutions of the three sets of algebraic equations f(E) + i(i + √ 3)δ1(E) − i � −i + √ 3 � g(E) δ1(E) = 24p(p + 2)E, η2(E) − 1 24 � � �12 + ϵ √ 6 � � � �f(E) E − i � −i + √ 3 � δ1(E) E + i � i + √ 3 � g(E) Eδ1(E) + 24 � � � + p + 1 = 0, where again ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η3(E) = 1 24 � 12 − √ 6 � f(E) E − (1+i √ 3)δ1(E) E + (−1+i √ 3)g(E) Eδ1(E) + 24 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This η yields the algebraic equa- tion whose real solutions gives other possible energies of the system, f(E) − (1 + i √ 3)δ1(E) + (−1 + i √ 3)g(E) δ1(E) = 24p(p + 2)E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is in general very difficult to solve the above algebraic equations for E analytically due to their complicated forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' To demonstrate the existence of real solutions to the above algebraic equations, we have a closer look at cases of restricted model parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' As an example, we consider b1 = b2 = b3 = h for any h ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this case the structure function reduces to Φ(III) 2 (z, η) = � z + η − 1 2 �2 � �z + η − 1 8 � �4 − � 2h2(E) − 2h1(E) E � � � � � �z + η − 1 8 � �4 + � 2h2(E) − 2h1(E) E � � � � � �z + η − 1 8 � �4 − � 2h2(E) + 2h1(E) E � � � � � �z + η − 1 8 � �4 + � 2h2(E) + 2h1(E) E � � � � , where h1(E) = � h4 + 16h3E + 8h2E(12E + 1) + 64hE2(4E − 15) + 16E2 (16E2 + 8E + 17), h2(E) = h2 − 24hE + 16E2 + 12E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3), we determine the constant η and the corresponding energies for the model parameters b1 = b2 = b3 = h as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 16 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η1 = 1 2 and E1,± = 4h2 + 12hp2 + 24hp + 15h + 8p4 + 32p3 + 42p2 + 20p + 1 ± m(h, p) 4 (4h + 4p2 + 8p + 3) , where m(h, p) = � (4h2 + 3h (4p2 + 8p + 5) + 8p4 + 32p3 + 42p2 + 20p + 1)2 − h2 (4h + 4p2 + 8p + 3)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is easy to check that m(p) is real for any p ∈ N+ if h > 0 and so E1,± give the energies of the system for the model parameters b1 = b2 = b3 = h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η2(E) = 1 8 � 4 + ϵ � 2h2(E)−2h1(E) E � with ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For this η value, the energies are E2,± = 8h2 + 6hp2 + 12hp + 12h + p4 + 4p3 + 3p2 − 2p − 4 ± n(h, p) 8(4h + p(p + 2)) , where n(h, p) = � (8h2 + 6h (p2 + 2p + 2) + p4 + 4p3 + 3p2 − 2p − 4)2 − 4h2(4h + p(p + 2))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It can be checked that n(p) is real for h > 1 and so E2,± give the energies of the system for the model parameters b1 = b2 = b3 = h > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Other possible energies corresponding to η2(E) are E3,± = 1 8 � 4 � (1 − 4h)p(p + 2) − 2h + 4p2 + 8p + 3 ± l(h, p) � , where l(p, h) = 8p2� 4z(h, p)(16p + 7) − 4h (4z(h, p) + 20p2 + 40p + 3) + 16p4 + 64p3 + 80p + 22, z(p, h) = � (1 − 4h)p(p + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is easily seen that l(p) and z(p) are real for h < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Hence, E3,± are real for h < 0 and give the energies of the system for model parameters b1 = b2 = b3 = h < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 Potential V3(µ, ν) The constants of motion of the superintegrable system with potential V3 and the Hamiltonian ˆH = µ2∂2 µ−ν2∂2 ν (µ+ν)(2+µ−ν) + V3(µ, ν) are given by [14], A = −4µ2ν2 (∂µ + ∂ν)2 (µ + ν)2 − c2 µ − ν µν − c3 (µ − ν)2 µ2ν2 , B = ν2(µ + 2)µ∂2 ν − µ2(ν − 2)ν∂2 µ (µ + ν)(2 + µ − ν) − 4µ2ν2 (∂µ + ∂ν)2 (µ + ν)2 − c1µ2ν2 + c2µν + 2c3(1 + µ − ν) µν(2 + µ − ν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These integrals form the quadratic algebra with the commutation relations [A, B] = C, [A, C] = −2{A, B} − B − 2c1c2 + 4c2 ˆH, [B, C] = 2B2 − 8c3 ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The Casimir operator for the quadratic algebra is given by K3 = C2 + 2{A, B2} − 16c3 ˆHA + 5B2 + 4c2 � c1 − 2 ˆH � B, 17 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES which can be expressed in terms of the Hamiltonian as K3 = 16c3 ˆH2 + 4(c2 2 − 4c1c3) ˆH + 4c2 1c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It can be shown that A = (N + η)2 − 1 2, B = − c1c2 − 2c2 ˆH 2 � (N + η)2 − 1 4 � + b†ρ(N) + ρ(N)b, where η is a constant to be determined and ρ(N) = 1 3 · 220(N + η)(1 + N + η)(1 + 2(N + η))2 , convert the quadratic algebra into the deformed oscillator algebra with the structure function Φ(III) 3 (N, η) = −786432 � −c2 1 + 4c1 ˆH + ˆH (2N + 2η − 1)2 − 4 ˆH2� � c2 2 − c3(2N + 2η − 1)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By acting it on a Fock basis |z, E⟩, the structure function becomes Φ(III) 3 (z, η) = 12582912 c3 E � z + η − � 1 2 − c2 2√c3 �� � z + η − � 1 2 + c2 2√c3 �� � z + η − E − √ E(c1 − 2E) 2E � � z + η − E + √ E(c1 − 2E) 2E � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraint conditions (3), we obtain 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η(E) = E− √ E(c1−2E) 2E or η(E) = E+ √ E(c1−2E) 2E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For both cases, we have E = 1 8 � 4c1 + (p + 1)2 ± (p + 1) � 8c1 + (p + 1)2 � , which is real for c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This gives the energy spectrum of the system for any model parameters c1, c2, c3 with c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ηϵ = 1 2 � 1 + ϵ c2 √c3 � , where ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The corresponding energies are given by Eϵ = 1 8 � ��4c1 + � 2(p + 1) + ϵ c2 √c3 �2 ± � 2(p + 1) + ϵ c2 √c3 � � � � �8c1 + � 2(p + 1) + ϵ c2 √c3 �2 � �� , which is real for c1 > 0 and c3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 Potential V4(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ν) For a superintegrable system with the Hamiltonian ˆH = µ2∂2 µ−ν2∂2 ν (2+µ−ν)(µ+ν)+V4(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ν) associated to the potential V4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' the constants of motion are given by [14] A = ν2(µ + 2)µ∂2 ν − µ2(ν − 2)ν∂2 µ (µ + ν)(2 + µ − ν) − µν (d1(ν − 2) + d2(µ + 2) + 2d3(ν − µ + µν)) (µ + ν)(2 + µ − ν) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = 1 4µν(µ − ν + 2)(µ + ν)2 �� µ4(12ν3 − 12ν2 + ν + 1) + 2µ3ν − (ν − 1)µ2ν2� ∂2 µ + µν(µ − ν + 2) � µ2(12ν2 + 1) + 2µν + ν2� ∂µ∂ν +ν2 � µ3(12ν2 − 1) + µ2(12ν2 − 1) + µ(ν − 2)ν − ν2� ∂2 ν � − (µ − ν) �(µ − ν)(d1µ + d2ν) − 2d3(µ2 + ν2 + µν(2 + µ − ν)) � 4µν(µ + ν)(2 + µ − ν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 18 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES They satisfy the quadratic algebra relations [A, B] = C, [B, C] = −2B2 + 2 ˆHB − d2 3 2 , [A, C] = 2{A, B} − 2 ˆHA − B + (d1 + d2 + 1 2) ˆH − d1d2 2 − 2 ˆH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By a direct calculation, we find the Casimir operator of this algebra K4 = C2 − 2{A, B2} + 5B2 + 2 ˆH{A, B} − d2 3A + � 4 ˆH − (2d1 + 2d2 + 5) ˆH + d1d2 � B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By means of the differential operator representation of A and B above, the Casimir operator K4 can be expressed in terms of ˆH as K4 = 4 ˆH3 − (2d1 + 2d2 + 1) ˆH2 + � (d1 + d2)2 4 + d3(d2 − d1) � ˆH − d3(d3 − d2 1 + d2 2) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We can show that A =(N + η)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = ˆH 2 − −4 ˆH + 8(d1 + d2 + 1 2) ˆH − 4d1d2 32 � (N + η)2 − 1 4 � + ρ(N)b† + bρ(N) with ρ(N) = 1 3 · 220(N + η)(1 + N + η)(1 + 2(N + η))2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' give a realization of the quadratic algebra in terms of the deformed oscillator algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' with the structure function given by Φ(III) 4 (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) =16384(1 − 2(N + η))2 � 3 � d3 � −d2 1 + d2 2 + d3 � + 4d3 ˆH(d1 − d2) +4 ˆH2(2d1 + 2d2 + 1) − ˆH(d1 + d2)2 − 16 ˆH3� + 6d1 ˆH(d2 − 2 ˆH) −4 ˆH2(3d2 − 6 ˆH + 2) + � ˆH2 − d2 3 � � 12(N + η)2 − 12(N + η) − 1 � − 7d2 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Here η is a constant parameter to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Acting on the Fock states |z, E⟩, the structure function is factorized as Φ(III) 3 (z, η) = � z + η − 1 2 �2 � z + η − � 1 2 − γ1(E) 2 �d2 3 − E2� �� � z + η − � 1 2 + γ1(E) 2 �d2 3 − E2� �� , where γ1(E) = � d2 3 − E2 × � −d2 1(d3 + E) + 4d1E(d3 + E) + d2 2(d3 − E) + 4d2E(E − d3) − 8E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' From the constraints (3), we determine the constant η and the energy spectrum E of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We list the results as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 1: η(E) = 1 2 − γ1(E) 2(d2 3−E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Corresponding to this η value, we have either � −d2 1(d3 + E) + 4d1E(d3 + E) + d2 2(d3 − E) + 4d2E(E − d3) − 8E3 = (p + 1) ��d2 3 − E2� (14) or � −d2 1(d3 + E) + 4d1E(d3 + E) + d2 2(d3 − E) + 4d2E(E − d3) − 8E3 = 2(p + 1) ��d2 3 − E2�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (15) 19 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES Notice that obviously the solution space of the algebraic equation (15) is subspace of that of (14) and so the energy spectrum of the system corresponding to η1(E) is given by solutions to (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 2: η = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this case, E satisfies the same algebraic equation as (15) and so do not give new energies of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 3: η(E) = � 1 2 + γ1(E) 2(d2 3−E2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This η value give the same equations for E as those in Case 1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Due to the complexity of the algebraic equations, it is hard to see whether or not they lead to real energies E for general model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' However, we can show that when the model parameter d3 = 0, the structure function reduces to Φ(III) 3 (z, η) = � z + η − 1 2 �2 � z + η − 1 2E � E − � E �d2 1 − 4d1E + d2 2 − 4d2E + 8E2��� � z + η − 1 2E � E + � E �d2 1 − 4d1E + d2 2 − 4d2E + 8E2��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this case, by imposing the constraints (3) we obtain the parameter η and the energies of the system with model parameter d3 = 0, η−(E) = 1 2E � E − � E �d2 1 − 4d1E + d2 2 − 4d2E + 8E2�� , E± = 1 16 � 4d1 + 4d2 + (p + 1)2 ± � (p + 1)2 ((p + 1)2 + 8(d1 + d2)) − 16 (d1 − d2)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Other η values from the constraints give rise to same energies as E± above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is clear that both E± are real for d1 = d2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' So E± give the energy spectrum of the system for model parameters d1 = d2 > 0, d3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The corresponding structure function of the p + 1)-dimensional unirreps is Φ(III)E±(z) = z(z − p − 1) � z − 1 2E± � E± �d2 1 − 4d1E± + d2 2 − 4d2E± + 8E2 ± � �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 Darboux Space IV In Darboux space IV, there are 3 different potentials in the separable coordinates (µ, ν), (u, v) and (ω, ϕ): V1(µ, ν) = −sin2(2µ)(4a1 exp(2ν) + 4a2 csc2(2µ) + 4a3 exp(4ν)) 2 cos 2µ + a4 , V2(u, v) = − sin2(2u)( b2 sinh2 v + b3 cosh2 v) + b1 2 cos 2u + b4 , V3(ω, ϕ) = c1 cos2 ϕ + c2 cosh2 ω + c3 � 1 sin2 ϕ − 1 sinh2 ω � c4+2 sinh2(2ω) + c4−2 sin2(2ϕ) , where ai, bi ci are real model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 Potential V1(µ, ν) The integrals of motion of the superintegrable system in Darboux space IV with potential V1 and the Hamiltonian ˆH = − 4µ2ν2 (a4+2)µ2+(a4−2)ν2 + V1(µ, ν) are A =µ2∂2 µ + 2µν∂µ∂ν + ν2∂2 ν + µ∂µ + ν∂ν + a1(µ2 + ν2) + a3(µ2 + ν2)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = 4(a4 + 2)µ2∂2 µ − 4(a4 − 2)ν2∂2 ν (a4 + 2)µ2 + (a4 − 2)ν2 + 2a1 �(a4 + 2)µ2 − (a4 − 2)ν2� + 4a3 �(a4 + 2)µ4 − (a4 − 2)ν4� + 16a2 (a4 + 2)µ2 + (a4 − 2)ν2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 20 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES They form the quadratic algebra with the commutation relations given by [14] [A, B] = C, [A, C] = 8{A, B}a − 16B + 32a1 ˆH, [B, C] = −8B2 + 256a3A + 128 a3a4 ˆH + 32(a2 1 + 4a3 + 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By a direct calculation, we find that the Casimir operator is K1 = C2 − 8{A, B2} + 256 a3A2 + 80B2 + � 256 a3a4 ˆH + 64(16a2a3 + a2 1 + 4a3) � A − 64 a1 ˆHB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' With the differential operator representation of A and B, the Casimir operator can be expressed in terms of ˆH as K1 = −256 a3 ˆH2 + 64 a4(4a3 − a2 1) ˆH + 128(a2 1 + 4a3 + 8a2a3 − 2a2 1a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After a long calculation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we find that the change of basis A = 4(N + η)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = − 128a1 ˆH 256 � (N + η)2 − 1 4 � + ρ(N)b† + bρ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where ρ(N) = 1 3 · 215(N + η)(1 + N + η)(1 + 2(N + η))2 maps the quadratic algebra to the deformed oscillator algebra with the structure function Φ(IV ) 1 (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = − 805306368 (2(N + η) − 1)2 × � 128(−2a2 1a2 + a2 1 + 8a2a3 + 4a3) + 64 ˆHa4(4a3 − a2 1) − 256a3 ˆH2� + 131072 � 12(N + η)2 − 12(N + η) − 1 � (2(N + η) − 1)2 × � 131072(a2 1 + 16a2a3 + 4a3) + 524288 a3a4 ˆH + 1048576 a3 � − 16384 (2(N + η) − 1)2 � −7340032(a2 1 + 16a2a3 + 4a3) + 29360128 a3a4 ˆH − 46137344 a3 � + 51539607552 a2 1 ˆH2 + 206158430208 a3 (2(N + η) − 3) (2(N + η) + 1) (2(N + η) − 1)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Acting on the Fock basis state |z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' E⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we find that the structure function has the factorization Φ(IV ) 1 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = � z + η − � 1 2 − ia1 4√a3 �� � z + η − � 1 2 + ia1 4√a3 �� � z + η − 1 2 √ 2 �√ 2 − � 1 − 4a2 − Ea4 − m1(E) �� � z + η − 1 2 √ 2 �√ 2 + � 1 − 4a2 − Ea4 − m1(E) �� � z + η − 1 2 √ 2 �√ 2 − � 1 − 4a2 − Ea4 + m1(E) �� � z + η − 1 2 √ 2 �√ 2 + � 1 − 4a2 − Ea4 + m1(E) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where m1(E) = � (4a2 + Ea4 + 1)2 − 4(4a2 + E(E + a4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3), we have 21 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η1(E) = 1 2 √ 2 �√ 2 − � 1 − 4a2 − Ea4 − m1(E) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This η value gives the following two sets of energies and corresponding structure functions E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ =1 2(p + 1) � −(p + 1) a4 + ϵ � (a2 4 − 4)(p + 1)2 + 16(1 − a2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Φ(IV ) E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ (z) =z(z − p − 1) � z − 1 2 √ 2 � 1 − 4a2 − E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) − ia1 4√a3 � � z − 1 2 √ 2 � 1 − 4a2 − E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1(E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) + ia1 4√a3 � � z − 1 2 √ 2 �� 1 − 4a2 − E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1(E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) − � 1 − 4a2 − E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 + m1(E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ �� � z − 1 2 √ 2 �� 1 − 4a2 + E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1(E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) + � 1 − 4a2 − E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 + m1(E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ = − 1 a4 + ϵ 4 � 4a2 + 4p2 + 8p + 3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' a4 ̸= ±4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ΦE2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ(z) =z(z − p − 1) � z − 1 2 √ 2 � 1 − 4a2 − E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) − ia1 4√a3 � � z − 1 2 √ 2 � 1 − 4a2 − E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1(E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) + ia1 4√a3 � � z − 1 √ 2 �� 1 − 4a2 − E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1(E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) �� � z − 1 2 √ 2 �� 1 − 4a2 + E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1(E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) + ϵ � 1 − 4a2 − E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 + m1(E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Notice that the energies E1,ϵ are real for a2 4 > 4, a2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η2(E) = 1 2 √ 2 �√ 2 − � 1 − 4a2 − Ea4 + m1(E) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The energies are the same as those given in case 1 above 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 Potential V2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' v) The constants of motion of the superintegrable system in Darboux space IV with the Hamiltonian ˆH = − sin2(2u)(∂2 v+∂2 u) 2 cos(2u)+b4 + V2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' v) are [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' A = e−2v ��e4v + 1 � (2b4 cos(2u) + 3 cos(4u) + 1) 2b4 + 4 cos(2u) ∂2 v − sin(2u) �e4v + 1 � sin(2u) b4 + 2 cos(2u) ∂2 u � + e−2v � sin(2u) � e4v + 1 � ∂u + sin(2u) � e4v − 1 � ∂u∂v + cos(2u) � e4v − 1 � ∂v � + 1 2 cos 2u + b4 � 2b1 cosh 2v + (b2 + b3)(4 − b2 4) + (cos 4u + 2b4 cos 2u + 3) � b2 sinh2 v − b3 cosh2 v �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = ∂2 v + b2 sinh2 v + b3 cosh2 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These integrals generates the quadratic algebra the commutation relations as follows [A, B] =C, [A, C] =8{A, B} + 16b4(b2 + b3)A − 16B + 32(b1 + b3) ˆH − 16b4(b2 + b3), [B, C] =8B2 + 96A2 + � 64b4 ˆH + (2b2 − 2b3 + b1 + 3) � A + 32 ˆH2 + 32b4(2b2 − 2b3 + 1) ˆH + 64b1(b2 − b3) − 8(b2 4 − 4)(b2 + b3)2 + 32(b1 + 2b2 − 2b3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 22 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES By a direct computation, we find the Casimir operator of the algbebra, K2 = C2 + 64A3 − 8{A, B2} − 16b4(b2 + b3){A, B} + 64 � b4 ˆH + 2b2 − 2b3 + b1 + 7 � A2 + � 160b4(b2 + b3) − 64(b2 + b3) ˆH � B − 64b4(2b3 − 2b2 − 1) ˆHA − 16 � (b2 4 − 4)(b2 + b3)2 + 8(b1 + 1)(b3 − b2) − 4b1 + 32 � A + 64 ˆH2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' With the differential realization of A and B, the Casimir K2 is expressible in terms of ˆH as follows K2 =128(b3 − b2 + 1) ˆH2 + 128b4(b2 − b3 + 1) ˆH + (128 − 80b2 4 − 64b1)(b2 + b3)2 − 128(b1 + 2)(b3 − b2 − 1) − 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After a long computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we find that the realization A = −4 � (N + η)2 − 1 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = b4(b2 + b3) 8 − −32 · 16b4(b2 + b3) − 256 · (b1 + 2b2 − 2b3) 4γ3 � (N + η)2 − 1 4 � + ρ(N)b† + bρ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where ρ(N) = 1 3 · 212 · (−8)8(N + η)(1 + N + η)(1 + 2(N + η))2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' converts the quadratic algebra into the deformed oscillator algebra with the structure function Φ(IV ) 2 (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) =268435456 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(2N + 2η − 1)2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='48(5a2 + 4b1 − 8)(b2 + b3)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+384 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(b1 + 2)(b2 − b3 + 1) + ˆH2(−b2 + b3 + 1) + ˆHb4(b2 − b3 + 1) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 64 (2N + 2η − 1)2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='12(N + η)2 − 12(N + η−) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b1(2b2 − 2b3 + 3) + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + 2b2(b3 + ˆHb4 + 3) + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 2b3 ˆHb4 − 6b3 + ˆH2 + 3 ˆHb4 + 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ (2N + 2η + 1)(2N + 2η − 1)6(b1 + 2b2 − 2b3 + ˆHb4 + 6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='986 b1(b2 − b3) + 448 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b1 + 2b2 − 2b3 + ˆHb4(2b2 − 2b3 + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 112 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(b2 + b3)2 + 448 ˆH2 + 96 b4(b2 + b3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ˆH(b1 + b3) − b4(b2 + b3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+704 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b1 + 2b2 − 2b3 + ˆHb4 + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 32b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4(b2 + b3)2 + 192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2N + 2η − 3 + ˆH2(b1 + b3)2 − (2N + 2η − 1)4 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4(N + η)2 − 4(N + η) − 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 23 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES Acting on Fock basis states |z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' E⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' the structure function is factorized as follows Φ(IV ) 2 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 2b2 + 2b3 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(1 − 4b2)(1 + 4b3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 2b2 + 2b3 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(1 − 4b2)(1 + 4b3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 2b2 + 2b3 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(1 − 4b2)(1 + 4b3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 2b2 + 2b3 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(1 − 4b2)(1 + 4b3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − b1 − Eb4 − m2(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − b1 − Eb4 − m2(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − b1 − Eb4 + m2(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − b1 − Eb4 + m2(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where m2(E) = � (1 + b1 + Eb4)2 − 4 (b1 + E2 + Eb4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraint conditions (3), we determine the parameter η and the corresponding energies of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We find Case 1: η1,± = 1 2 √ 2 �√ 2 − � 1 − 2b2 + 2b3 ± � (1 − 4b2)(1 + 4b3) � and the energies E satisfy 2 √ 2(p + 1) − n2,± = � 1 − b1 − Eb4 + m2(E), where n2,± = � 1 − 2b2 + 2b3 ± � (1 − 4b2)(1 + 4b3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Noticing (1 − 4b2)(1 + 4b3) = (1 − 2b2 + 2b3)2 − (2b2 + 2b3)2 ≤ (1 − 2b2 + 2b3)2, we conclude that both n2,± are real if b2 < 1 4, b3 > −1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Solving the algebraic equations give the energies of the system and the corresponding structure func- tions of the (p + 1)-dimensional unirreps of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We have E1±,ϵ = − b4 4 � 2 √ 2(p + 1) − n2,± �2 + ϵ � 2 √ 2(p + 1) − n2,± � � 8(1 − b1) + (b2 4 − 4) � 2 √ 2(p + 1) − n2,± �2, Φ(IV ) E1±,ϵ(z, η) =z (z − p − 1) � z − 1 √ 2n± � � z − 1 2 √ 2(n± − n∓) � � z − 1 2 √ 2(n± + n∓) � � z − 1 2 √ 2 � n± − � 1 − b1 − E1±,ϵb4 − m2(E1±,ϵ) �� � z − 1 2 √ 2 � n± + � 1 − b1 − E1±,ϵb4 − m2(E1±,ϵ) �� � z − 1 2 √ 2 � n± − � 1 − b1 − E1±,ϵb4 + m2(E1±,ϵ) �� , where ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The energies E1±,ϵ are real for the model parameters b2 < 1 4, b3 > −1 4, b1 < 1, b2 4 > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 24 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES Case 2: η1,± = 1 2 √ 2 �√ 2 + � 1 − 2b2 + 2b3 ± � (1 − 4b2)(1 + 4b3) � and the energies E satisfy 2 √ 2(p + 1) + n2,± = � 1 − b1 − Eb4 + m2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Solutions of the equations give the energies of the system and the corresponding structre functions of the (p + 1)-dimensional unirreps of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We have E2±,ϵ = − b4 4 � 2 √ 2(p + 1) + n2,± �2 + ϵ � 2 √ 2(p + 1) + n2,± � � 8(1 − b1) + (b2 4 − 4) � 2 √ 2(p + 1) + n2,± �2, Φ(IV ) E2±,ϵ(z, η) =z (z − p − 1) � z + 1 √ 2n± � � z + 1 2 √ 2(n± − n∓) � � z + 1 2 √ 2(n± + n∓) � � z + 1 2 √ 2 � n± − � 1 − b1 − E2±,ϵb4 − m2(E2±,ϵ) �� � z + 1 2 √ 2 � n± + � 1 − b1 − E2±,ϵb4 − m2(E2±,ϵ) �� � z + 1 2 √ 2 � n± + � 1 − b1 − E2±,ϵb4 + m2(E2±,ϵ) �� , where ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The energies E2±,ϵ are real for the model parameters b2 < 1 4, b3 > −1 4, b1 < 1, b2 4 > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 3: η3,−(E) = 1 2 √ 2 �√ 2 − � 1 − b1 − Eb4 + m2(E) � and √ 2(p + 1) = � 1 − b1 − Eb4 + m2(E) or 2 √ 2(p + 1) = � 1 − b1 − Eb4 + m2(E) + � 1 − b1 − Eb4 − m2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The first algebraic equation gives the energies E3,1 = p + 1 2 � −(p + 1)b4 ± � 4(1 − b1) + (b2 4 − 4)(p + 1)2 � , which is real for b1 < 1, b2 4 > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The corresponding structure function is Φ(IV ) E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) =z (z − p − 1) � z − 1 2 √ 2 �� 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 + m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) − n− �� � z − 1 2 √ 2 �� 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 + m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) + n− �� � z − 1 2 √ 2 �� 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 + m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) − n+ �� � z − 1 2 √ 2 �� 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 + m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) + n+ �� � z − 1 2 √ 2 �� 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 + m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) − � 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 − m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) �� � z − 1 2 √ 2 �� 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 + m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) + � 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 − m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The second algebraic equation yields the energies E3,2 = 1 2 − b4 � 4(p + 1)2 + b1 − 1 � , 25 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES which is well defined for the model parameter b4 ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The associated structure function is given by Φ(IV ) E3,2 (z, η) =z (z − p − 1) � z − 1 2 √ 2 �� 1 − b1 − E3,2b4 + m2(E3,2) − n− �� � z − 1 2 √ 2 �� 1 − b1 − E3,2b4 + m2(E3,2) + n− �� � z − 1 2 √ 2 �� 1 − b1 − E3,2b4 + m2(E3,2) − n+ �� � z − 1 2 √ 2 �� 1 − b1 − E3,2b4 + m2(E3,2) + n+ �� � z − 1 2 √ 2 �� 1 − b1 − E3,2b4 + m2(E3,2) − � 1 − b1 − E3,2b4 − m2(E3,2) �� � z − 1 √ 2 � 1 − b1 − E3,2b4 + m2(E3,2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 4: η4,+ = 1 2 √ 2 �√ 2 + � 1 − b1 − Eb4 − m2(E) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We have the algebraic equation 2 √ 2(p + 1) = � 1 − b1 − Eb4 + m2(E) − � 1 − b1 − Eb4 − m2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Solving, we obtain E4 = 1 2 + b4 � 4(p + 1)2 + b1 − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This gives the energy spectrum of the system for the model parameter b4 ̸= −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The corresponding structure function is Φ(IV ) E4 (z, ��) =z (z − p − 1) � z + 1 2 √ 2 �� 1 − b1 − E4b4 − m2(E4) − n− �� � z + 1 2 √ 2 �� 1 − b1 − E4b4 − m2(E4) + n− �� � z + 1 2 √ 2 �� 1 − b1 − E4b4 − m2(E4) − n+ �� � z + 1 2 √ 2 �� 1 − b1 − E4b4 − m2(E4) + n+ �� � z + 1 √ 2 � 1 − b1 − E4b4 − m2(E4) � � z + 1 2 √ 2 �� 1 − b1 − E4b4 − m2(E4) + � 1 − b1 − E4b4 + m2(E4) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 26 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 Potential V3(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ϕ) The constants of motion of the superintegrable system in Darboux space IV with the potential V3(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ϕ) are [14] A = − 2c4 ∂2 ϕ + ∂2 ω c4+2 sinh2(2ω) + c4−2 sin2(2ϕ) + (c4 + 2) sin2(2ϕ)∂2 ϕ − (c4 − 2) sinh2(2ω)∂2 ω (c4 + 2) sin2(2ϕ) + (c4 − 2) sinh2(2ω) + 1 c4+2 sinh2(2ω) + c4−2 sin2 ω � c4 + 2 sinh2(2ω) � c3 sin2 ϕ + c1 cos2 ϕ � + c4 − 2 sin2(2ω) � c3 sinh2 ω − c2 cosh2 ω �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='B =1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 sin(2ϕ) sinh(2ω) tan(ϕ − iω) tan(ϕ + iω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='cot(2ϕ) ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ω + coth(2ω) ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−i cos(2ϕ) sinh(2ω) sinh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�tan(ϕ − iω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='tan(ϕ + iω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='∂ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+i cosh(2ω) sin(2ϕ) sinh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�tan(ϕ − iω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='tan(ϕ + iω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='∂ϕ + 2 cosh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�tan(ϕ − iω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='tan(ϕ + iω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='c4+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='sinh2 2ω + c4−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='sin2 ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� c4 + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='sinh2 2ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�c1 cosh 2ω tan2 ϕ − c2 cos 2ϕ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 cos2 ϕ(sinh2 ω − sin2 ϕ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='sin2 ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ c4 − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='sin2 2ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�c2 cos 2ϕ tanh2 ω + c1 cosh 2ω − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 cosh2 ω(sinh2 ω − sin2 ϕ) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='sinh2 ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These integrals form the quadratic algebra with the following commutation relations [A, B] =C, [A, C] = − 8{A, B} − 16B − 16(c1 − c3)(c2 − c3), [B, C] = − 24A2 + 8B2 + 16 � 2c4 ˆH − 2c1 + 2c2 + 3 � A − 16 [(c4 + 2)c1 + (c4 − 2)c2 − c4 + 64c3] ˆH − 8(c2 4 − 4) ˆH2 − 8c2 1 − 8c2 2 + 16c2 3 + 32c1c2 + 48c3(c1 + c2) − 16(c1 − c2), which is the symmetry algebra of the superintegrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We can calculate the Casimir operator of the algebra K3 =C2 − 16A3 + 8{A, B2} − 16(2c2 − 2c1 − 7)A2 + 80B2 − 16 � c2 4 − 4 + 2(c4 + 2)c1 − 2(c4 − 2)c2 + 8c3 + 2c4 � ˆH + 16 � c2 1 + c2 2 − 2c2 3 − 6c3(c1 + c2) − 4c1c2 + 2c1 − 2c2 − 8 � A + 32(c2 − c3)(c1 − c3)B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We can show that in terms of the Hamiltonian the Casimir K3 takes the simple form K3 =16(c2 4 − 4) ˆH2 − 16 � (c4 + 2)((c1 − c3)2 − 2c1) + (c4 − 2)((c2 − c3)2 + 2c2) − 8c3 − 4c4 � ˆH − 32(c1 − c2)(3c2 3 − c1c2 − c3(c1 + c2)) + 32(c2 1 + c2 2 − 4c3(c1 + c2) − 2c1c2 + 2c1 − 2c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After a long computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we find that the realization A(N) = −4(N + η)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = −(c1 − c3)(c2 − c3) 4 � (N + η)2 − 1 4 � + ρ(N)b + b†ρ(N) with ρ(N) = 1 3 · 212 · (−8)8(N + η)(1 + N + η)(1 + 2(N + η))2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 27 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES changes the quadratic algebra to the deformed oscillator algebra with the structure function Φ(IV ) 3 (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) =268435456 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='12N 2 + 12N(2η − 1) + 12η2 − 12η − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(2N + 2η − 1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 + c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−4c2 − 6c3 + 2 ˆHc4 + 4 ˆH − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−6c3 − 2 ˆHc4 + 4 ˆH + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 128c3 ˆH + ˆH2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 − 4 ˆH2 + 6 ˆHc4 + 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 16 (2N + 2η − 1)2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='7c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 2c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='14c2 + 21c3 − 7 ˆHc4 − 14 ˆH + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 7c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−42c3 − 14 ˆH(c4 − 2) + 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 14c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 896c3 ˆH + 7 ˆH2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 − 28 ˆH2 + 36 ˆHc4 + 36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 3 (2N + 2η − 1)2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='352 ˆH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−c1 + (c4 − 2)(c2 − c3)2 + 2c2(c4 − 2) − c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 8c3 − 4c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+32(c1 − c2) (c1(c2 + c3) + c3(c2 − 3c3)) − 16 ˆH2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 48(c1 − c3)2(c2 − c3)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 96 (2N + 2η − 3)(2N + 2η + 1)(2N + 2η − 1)4(c1 − c2 − ˆHc4 − 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+48 (2N + 2η − 1)4 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(2N + 2η − 1)4 − 8(2N + 2η − 1)2 + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure function is a polynomial of degree 8 in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Acting on the Fock basis states |z, E⟩, it becomes a polynomial of degree 8 in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In order to determine the energy spectrum of the superintegrable system, we have to find the finite-dimensional unirreps of the deformed oscillator algebra by solving the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This requires the factorization of the structure function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' However, it turns out to be very difficult to factorize the structure function for general model parameters ci (even using symbolic computation softwares such as Mathematica).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In the following we restrict our attention to special model parameters and present analytic and closed-form results for c1 = c2 = 1, c3 = c4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we find that the structure function factorizes as Φ(IV ) 3 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 − 12E + 5 − 2E + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 − 12E + 5 − 2E + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 − 12E + 5 − 2E + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 − 12E + 5 − 2E + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 − 4E − 3 + 2E − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 − 4E − 3 + 2E − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3), we determine the constant η and obtain the energies of the system and the structure function of the symmetry algebra for the model parameters c1 = c2 = 1, c3 = c4 = 0 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The constant η is ηa(E) = 1 2 √ 2 �√ 2 − � − √ 4E2 − 12E + 5 − 2E + 3 � and the energies are given by the equation 2 √ 2(p + 1) = �� 4E2 − 12E + 5 − 2E + 3 + � − � 4E2 − 12E + 5 − 2E + 3 =⇒ Ea = −2(p + 1)2 + 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 28 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES The associated structure function is Φ(IV ) Ea (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = z (z − p − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2a − 12E2a + 5 − 2Ea + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2a �� 12Ea + 5 − 2Ea + 3 − ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b − 12Eb + 5 − 2Eb + 3 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b − 4Eb − 3 + 2Eb − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The constant η is η2c(E) = 1 2 √ 2 �√ 2 + � − √ 4E2 − 12E + 5 − 2E + 3 � and the energy E satisfies 2 √ 2(p + 1) = �� 4E2 − 12E + 5 − 2E + 3 − � − � 4E2 − 12E + 5 − 2E + 3 =⇒ Ec = −2(p + 1)2 + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure function is given by Φ(IV ) Ec (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = z (z − p − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + 1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2c − 12Ec + 5 − 2Ec + 3 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2c − 4Ec − 3 + 2Ec − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 29 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The constant η is ηd(E) = 1 2 √ 2 �√ 2 − � − √ 4E2 − 4E − 3 + 2E − 1 � and the energies are 2 √ 2(p + 1) = �� 4E2 − 4E − 3 + 2E − 1 + � − � 4E2 − 4E − 3 + 2E − 1 =⇒ Ed = 2(p + 1)2 − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The corresponding structure function reads Φ(IV ) d (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = z (z − p − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3 New superintegrable systems in 2D Darboux spaces In this section, we investigate superintegrable systems in 2D Darboux spaces with linear and quadratic or quintic integrals of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We will first construct generic cubic and quintic algebras and derive their Casimir operators and realizations in terms of the deformed oscillator algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We will then present examples of new superintegrable systems in 2D Darboux spaces with cubic symmetry algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 Generic cubic and quintic algebras generated by linear, quadratic or quintic integrals We start with the construction of generic cubic and quintic algebras with structure coefficients involving the Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Let ˆX1, ˆY1 be linear integrals, and let ˆX2, ˆY2 be quadratic and cubic integrals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' That is, deg ˆX1 = 1 = deg ˆY1, deg ˆX2 = 2, deg ˆY2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We define the operators ˆF and ˆG by ˆF = [ ˆX1, ˆX2] and ˆG = [ ˆY1, ˆY2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Then deg ˆF = deg ˆX1 + deg ˆX2 − 1 = 2 and deg ˆG = deg ˆY1 + deg ˆY2 − 1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By analysing the degrees of the integrals and applying the Jacobi identity constraint [34], we obtain the following generic cubic and quintic algebras Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Integrals { ˆX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆF} satisfy the cubic commutation relations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [ ˆX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2] = ˆF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [ ˆX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆF] =u1 ˆX2 1 + u2 ˆX1 + u3 ˆX2 + u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [ ˆX2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆF] =v1 ˆX3 1 + v2 ˆX2 1 + v3 ˆX1 − u2 ˆX2 − u1{ ˆX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2} + v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (16) and integrals { ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆG} form the following quintic commutation relations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [ ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2] = ˆG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [ ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆK] =α ˆY 3 1 + β ˆY 2 1 + δ ˆY1 + ϵ ˆY2 + ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [ ˆY2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆK] =a ˆY 5 1 + b ˆY 4 1 + c ˆY 3 1 + d ˆY 2 1 + e ˆY1 + 1 2 (α ϵ − 2 δ) ˆY2 − 3 2α{ ˆY 2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} − β{ ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} + z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (17) where uj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' , α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' , z are polynomials of the Hamiltonian ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Moreover, the coefficients v1 in (16)and a in (17) are not zero polynomials of ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The proof of this proposition is a short and straightforward computation from the Jacobi identity requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For the polynomials on both sides of the commutation relations (16) and (17) to have the same degree, we must have that v1, v2, u1, u2, u3, α, β, ϵ, a, b are constants and u = u(0) + u(1) ˆH, v3 = v(0) 3 + v(1) 3 ˆH, v = v(0) + v(1) ˆH, δ = δ(0) + δ(1) ˆH, ζ = ζ(0) + ζ(1) ˆH, c = c(0) + c(1) ˆH, d = d(0) + d(1) ˆH, e = e(0) + e(1) ˆH + e(2) ˆH2, z = z(0) + z(1) ˆH + z(2) ˆH2, 31 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES where u(0), u(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' , are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We now construct the Casimir operators for both polynomial algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We have Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The Casimir operators C(3) and C(5) for the cubic and quintic algebras are respectively given by C(3) = ˆF 2 − u1{ ˆX2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2} − u2{ ˆX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2} + v1 2 ˆX4 1 + 2 3v2 ˆX3 1 + � v3 + u2 1 � ˆX2 1 + (u1u2 + 2v) ˆX1 − 2u ˆX2 − u3 ˆX2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' C(5) = ˆG2 − α{ ˆY 3 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} + β{ ˆY 2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} − δ{ ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} − ϵ ˆY 2 2 − 2ζ ˆY2 + a 3 ˆY 6 1 + 2 5b ˆY 5 1 + 1 2 � c + 5 3aϵ + 3αδ � ˆY 4 1 + � 2β(δ + 3α) + 2 5ϵb − 2d � ˆY 3 1 + �1 6 (5 − 6a) ϵ2 + e + 1 2ϵc + β2 − 3 4α (αϵ − 2δ) � ˆY 2 1 + � 2z + βδ + βϵ(α + δ) − 1 5bϵ2 − ϵd � ˆY1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By analysinf the degrees of the integrals in the algebras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we see that the Casimir operators C(3) and C(5) of the cubic and quintic algebras have the following general form C(3) = ˆF 2 + w1{ ˆX2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2} + w2{ ˆX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2 2} + w3{ ˆX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2} + w4 ˆX4 1 + w5 ˆX3 1 + w6 ˆX2 1 + w7 ˆX1 + w8 ˆX2 + w9 ˆX2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' C(5) = ˆG2 + ω1{ ˆY 3 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} + ω2{ ˆY 2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} + ω3{ ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY 2 2 } + ω4{ ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} + ω5 ˆY 2 2 + ω6 ˆY2 + ω7 ˆY 6 1 + ω8 ˆY 5 1 + ω9 ˆY 4 1 + ω10 ˆY 3 1 + ω11 ˆY 2 1 + ω12 ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where wj and ωj are coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Now using the quadratic commutation relations (16), we have [C(3), ˆX1] = − (u1 + w1){ ˆF, ˆX2 1} − (w3 + u2 + w2u1){ ˆF, ˆX1} − (w9 + u3){ ˆF, ˆX2} − (2u + w8 + w2u2) ˆF − w2{ ˆF, { ˆX1, ˆX2}} + ˆX1(w1){ ˆX2 1, ˆX2} + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' + ˆX1(w9) ˆX2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Setting the coefficients to be zero gives w1 = −u1, w2 = 0, w3 = −u2, w8 = −2u, w9 = −u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Similarly, from [C(3), ˆX2] = 0,, we have 0 =(2w4 − v1){ ˆF, ˆX3 1} + (3w5 2 − v2){ ˆF, ˆX2 1} + (w6 − v3 − u2 1){ ˆF, ˆX1} + (w7 − u1u2 − 2v) ˆF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This gives w4 = v1 2 , w5 = 2v2 3 , w6 = v3 + u2 1, w7 = u1u2 + 2v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This finishes the proof for C(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The derivation of C(5) is slightly more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We express [C(5), ˆY1] and [C(5), ˆY2] in terms of { ˆG, ˆY n 1 } and { ˆG, { ˆY1, ˆY2}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By [34, Lemma 2] and quintic commutation relations, we have [C(5), ˆY1] = − (α + ω1){ ˆG, ˆY 3 1 } − � β + ω2 − 3αω3 2 � { ˆG, ˆY 2 1 } − (δ + ω3β + ω4){ ˆG, ˆY1} − (ϵ + ω5){ ˆG, ˆY2} − ω3{ ˆG, { ˆY1, ˆY2}} + ��αϵ 2 − δ � ω3 − 2ζ − ω6 � ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Setting the coefficients of { ˆK, { ˆY1, ˆY2}} and { ˆK, ˆY2} to be zero, we obtain that ω3 = 0 and ω5 = −ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Then [C(5), ˆY1] is reduced to the form [C(5), ˆY1] =(α − ω1){ ˆG, ˆY 3 1 } + (β − ω2){ ˆG, ˆY 2 1 } + [δ − ω4)]{ ˆG, ˆY1} + (2ζ − ω6) ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' From [C(5), ˆY1] = 0 it follows that the coefficients of { ˆG, ˆY l 1} are zero for all 1 ≤ n ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Thus ω1 = −α, ω2 = −β, ω4 = −δ and ω6 = −2ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 32 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES Similarly, after some manipulations we find [C(5), ˆY2] =(3ω7 − a){ ˆG, ˆY 5 1 } + �5ω8 2 − b � { ˆG, ˆY 4 1 } − (c + 3αδ + 5ϵω7 − 2ω9){ ˆG, ˆY 3 1 } + �3α 2 �αϵ 2 − δ � − β2 + 3αδϵ 2 + 3ϵ2ω7 − e − ϵω9 + ω11 � { ˆG, ˆY1} − � βδ + ϵω8 2 − 1 2ω10 − 3αβ + d � { ˆG, ˆY 2 1 } + � β �αϵ 2 − δ � − ϵω10 2 + ϵ2ω8 + 3αβϵ 2 + (ω12 − 2z) � ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It follows from [C(5), ˆY2] = 0 that ω7 = a 3, ω8 = 2b 5 , ω9 = 1 2 � c + 5ϵ 3 + 3αδ � , ω10 = 2(β(δ + 3α) + ϵb 5 − d) ω11 = �5 6 − a � ϵ2 + e + ϵc 2 + β2 − 3α 2 �αϵ 2 − δ � , ω12 = 2z + βδ + βϵ(α + δ) − bϵ2 5 − ϵd as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we now construct realizations of these algebras in terms of the deformed oscillator algebras (1) and determine their structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After long computations, we obtain the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The realization ˆX1 =√u3 (N + η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2 = − u3 (N + η)2 − u2 √u3 (N + η) + b† + b − u u3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (18) where η is a constant parameter to be determined,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' changes the cubic algebra (16) to the deformed oscillator algebra (1) with the structure function given by Φ(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = 1 1 − 2u3 � C(3) − u2 u3 + uu2 √u3 + √u3v + (N + η)2 � −2uu1 + 2u1u2 √u3 − u2 2 + (u2 + v2)u3/2 3 − u3 � + (N + η) � 2uu1 − 2uu2 √u3 − √u3(u1u2 + 2v) + u2 2 + u3v3 � +(N + η)3 � −2u1u2 √u3 + 2u1u2 3 − 2 3v2u3/2 3 + v1u2 3 � + (N + η)4 � −2u1u2 3 + u3 3 − 1 2v1u2 3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Note that Φ is a quartic polynomial of the number operator N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The transformation ˆY1 =√ϵ(N + η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2 = − α√ϵ(N + η)3 − β(N + η)2 − δ √ϵ(N + η) + b† + b − ζ ϵ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (19) where η is a constant parameter to be determined,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' maps the quintic algebra (17) to the deformed oscillator algebra with the structure function Φ(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4ϵC(5) + (N + η)6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�aϵ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='12 − 3α2ϵ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ (N + η)5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�3α2ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2αβϵ5/2 − aϵ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='10bϵ7/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ (N + η)4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4αβ√ϵ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='8ϵ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3αδ + 5aϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2αδϵ2 + β2ϵ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4bϵ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ (N + η)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4α(2αϵ − δ) − 3αδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2αζϵ3/2 − β2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2βδϵ3/2 − cϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4ω10ϵ5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ (N + η)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�β(2αϵ − δ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4√ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 3αζ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4√ϵ − βδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2√ϵ + βζϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ δ2ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 − d√ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ ω11ϵ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ (N + η) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�δ(2αϵ − δ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− βζ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2ϵ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2δζ√ϵ − e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4ω12ϵ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 + ζ(2αϵ − δ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4ϵ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure function Φ(N) is a polynomial of N of degree 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In the next subsection, we will present new superintegrable systems in 2D Darboux spaces with cubic symmetry algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 33 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 Superintegrable systems in 2D Darboux spaces with cubic symmetry algebras In this subsection we obtain potentials in the 2D Darboux spaces which can be added to the Hamiltonians of the free superintegrable systems studied in [11] and preserve their superintegrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The free systems have only kinetic terms and possess linear and quadratic integrals of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We will determine the integrals corresponding to the superintegrable systems with potnetials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 Darboux space I The Hamiltonian of the free system in Darboux space I with separable local coordinates (x, y) studied in [11] has the form H1 = ϕ1(x)(∂2 x + ∂2 y), where ϕ1(x) = 1 αx+β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This system has linear integral X1 = ∂y and quadratic integral given by X2 = y∂x∂y − x∂2 y + 1 2∂x − 1 4αy2ϕ1(x)(∂2 x + ∂2 y), where α is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We seek new superintegrable system in Darboux space I with Hamiltonian ˆH1 = H1 + V1(x, y), where V1(x, y) is potential function, which preserves the separability of the coordinates and the superintegrability of the original system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Without loss of generality, we assume that the local separable coordinates (x, y) is an orthogonal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After some computations, we find the allowed potential V1 and the corresponding integrals ˆX1, ˆX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The results are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆH1 = ϕ1(x)(∂2 x + ∂2 y) + c1ϕ1(x), ˆX1 = ∂y, ˆX2 = y∂x∂y − x∂2 y + 1 2∂x − 1 4αy2ϕ1(x)(∂2 x + ∂2 y) − 1 4c1αϕ1(x)y2 where c1 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By a direct calculation, we can show that the integrals ˆX1, ˆX2 form the cubic algebra, [ ˆX1, ˆX2] = ˆF, [ ˆX1, ˆF] = α 2 ˆH1, [ ˆX2, ˆF] = −2X3 1 + α ˆH1X1 − c1X1, (20) where explicitly ˆF = ∂x∂y − 1 2αyϕ1(x) � ∂2 x + ∂2 y � + 1 2c1αϕ1(x)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This cubic algebra is a special case of (16) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 with v1 = −2, u1 = u2 = u3 = v2 = v = 0, u = α 2 ˆH1, v3 = β ˆH1 − c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Then it follows that its Casimir operator is C(3) = ˆF 2 − X4 1 − α ˆH1 ˆX2 + (β ˆH1 − c1)X2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Since u3 = 0 it follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 that this cubic algebra does not have realization in terms of the deformed oscillator algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 Darboux space II The Hamiltonian of the free superintegrable system in 2D Darboux space II is H2 = ϕ2(x) � ∂2 x + ∂2 y � , where ϕ2(x) = x2 a2−a1x2 , a1, a2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The system possesses the following linear and quadratic integrals of motion, X1 = ∂y, X2 = 2xy∂x∂y + (y2 − x2)∂2 y + x∂x + y∂y + a1y2H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It can be shown that we can add the potential V2(x, y) = c2 ϕ2(x), where c2 is a real constant, to the free Hamiltonian such that ˆH2 = ϕ2(x) � ∂2 x + ∂2 y � + c2 ϕ2(x) is separable and superintegrable in the 2D Darboux space II, with integrals of motion given by ˆX1 = ∂y, ˆX2 = 2xy∂x∂y + (y2 − x2 + 1)∂2 y + x∂x + y∂y + a1y2H2 + a2c2y2 a2 − a1x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 34 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES By a direct computation, we find that these integrals obey the cubic commutation relations [ ˆX1, ˆX2] = ˆF, [ ˆX1, ˆF] = 2a1 ˆH2 + 2 ˆX2 1 + 2c2, [ ˆX2, ˆF] = 4 ˆX3 1 − 2{ ˆX1, ˆX2} + (2c2 + 1 − 2a2 ˆH2)X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (21) The Casimir operator of this cubic algeba is given by C(3) = ˆF 2 − 2{X2 1, ˆX2}a + 2 ˆX4 1 + (c2 + 5 − 2a2 ˆH2)X2 1 − 4(a1 ˆH2 + c2) ˆX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4, we again find that the cubic algebra has no realization in terms of the deformed oscillator algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 Darboux space III In the 2D Darboux space III, the free superintegrable system Hamiltonian and its constants of motion in the separable local coordinates (u, v) are given by H3 = ϕ3(v)(∂2 u + ∂2 v), X1 = ∂u, X2 = 1 2e−v � cos u(2∂2 u + ∂v) + sin u (2∂u∂v − ∂u) � + α cos uH3, where ϕ3(v) = e−v βev−2α with α, β being real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We seek potential of the form V3(u, v) = ϕ3(v)(f3(u) + g3(v)) such that system in Darboux space III with this potential is superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We thus expect that ˆH3 = H3 + V3(u, v) possesses linear and quadratic integrals of the form, ˆX1 = X1, ˆX2 = X2 + f3(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After some manipulations, we find that V3(u, v) = c3 ϕ3(v) and f3(u, v) = c3 βev cos(u) 2βev−4α , where c3 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' That is, we obtain the superintegrable system in Darboux space III with Hamiltonian and integrals given by ��H3 = ϕ3(v)(∂2 u + ∂2 v) + c3 βev − 2α, ˆX1 = ∂u, ˆX2 = 1 2 exp(−v) � cos u(2∂2 u + ∂v) + sin u(2∂u∂v − ∂u) � + α cos uH3 + c3 βev cos(u) 2βev − 4α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These integrals form the following algebra, [ ˆX1, ˆX2] = ˆF, [ ˆX1, ˆF] = − ˆX2, [ ˆX2, ˆF] = −β ˆH3 ˆX1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (22) The Casimir operator of this algebra is given by C(3) = ˆF 2−β ˆH3X2 1+ ˆX2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is interesting that the algebra generated by the above linear and quadratic integrals in the Darboux space III is “linear” in the generators (though with coefficient involving the Hamiltonian ˆH3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 Darboux space IV In terms of separable local coordinates (u, v), the Hamiltonian of the free superintegrable system in 2D Darboux space IV is H4 = ϕ4(u) � ∂2 u + ∂2 v � , where ϕ4(u) = sin2 u β−2α cos u α, β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The system possesses the following linear and quadratic integrals of motion, X1 = ∂v, X2 = exp(v) 2 � cos u(2∂2 v − ∂v) − sin u(2∂u∂v − ∂v) − 2αH4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By analysis similar to previous cases, we find that the system with the Hamiltonian ˆH4 = ϕ4(u) � ∂2 u + ∂2 v � + c4 β − 2α cos u, where c4 is a constant, is superintegrable with linear and quadratic integrals given by ˆX1 = ∂v, ˆX2 = exp(v) 2 � cos u(2∂2 v − ∂v) − sin u(2∂u∂v − ∂v) − 2αH4 � + 4c4e−v β − 2α cos u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These integrals form the cubic algebra, [X1, ˆX2] = ˆF, [X1, ˆF] = − ˆX2, [ ˆX2, ˆF] = 4X3 1 − 2β ˆH4X1 + 1 2 ˆX1 − 2αc4X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (23) 35 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES The Casimir operator of the algebra is C(3) = −1 2{ ˆX2, ˆF}a + β ˆH4X2 1 + X4 1 + (5 + 4c4)X2 1, which can be expressed as C(3) = ˆH2 4 + β ˆH4 + 4c4 in terms of the Hamiltonian ˆH4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Through the change of basis, X1 = (N + η), X2 = −(N + η)2 + b† + b the cubic algebra relations become those of the deformed oscillator algebra with structure function Φ(N, η) = (N + η)4 − 4(+N + η)3 + (N + η)2 − (N + η) � −2αc4 − 2βE + 1 2 � − 4c4 − E2 − βE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Here η is a constant which can be determined from the constraints on the structure function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 4 Conclusions We have presented a genuine algebraic analysis for the superintegrable systems in 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The main results in this paper are following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The first main result is the construction of the Casimir operators, deformed oscillator algebra realizations and finite-dimensional unirreps for all the 12 distinct quadratic algebras underlying the 12 superintegrable systems found in the classification of [14][15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This allows us to give an algebraic derivation for the energy spectrum of the 12 existing classes of superintegrable systems with quadratic integrals in the 2D Darboux spaces and the determination for the structure functions of the finite-dimensional unitary irreducible representations of the deformed oscillator algebras (corresponding to the quadratic algebras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' As our results demonstrate, superintegrable systems in curved (Darboux) spaces have much richer structures than those in flat spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For instance, the structures of energies of the systems and structure functions of the associated deformed oscillator algebras can be very complicated in the Darboux spaces, and in some cases we have to restrict the model parameter spaces in order to find explicit analytic and closed form solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Another main result of the paper is the construction of generic cubic and quintic algebras, generated by first, quadratic and cubic integrals, their Casimir operators and deformed oscillator algebra realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' As examples of applications, we obtain four classes of new superintegrable systems with non-trivial potentials and with linear and quadratic integrals in the 2D Darboux spaces, three of which have cubic algebras as their symmetry algebras.' metadata={'source': 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