Models of Quadratic Algebras Generated by Superintegrable Systems in 2D

In this paper, we consider operator realizations of quadratic algebras generated by second-order superintegrable systems in 2D. At least one such realization is given for each set of Stäckel equivalent systems for both degenerate and nondegenerate systems. In almost all cases, the models can be used...

Повний опис

Збережено в:

Бібліографічні деталі
Дата:	2011
Автор:	Post, S.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2011
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/146853
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Models of Quadratic Algebras Generated by Superintegrable Systems in 2D / S. Post // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 27 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-146853
record_format	dspace
spelling	irk-123456789-1468532019-02-12T01:24:33Z Models of Quadratic Algebras Generated by Superintegrable Systems in 2D Post, S. In this paper, we consider operator realizations of quadratic algebras generated by second-order superintegrable systems in 2D. At least one such realization is given for each set of Stäckel equivalent systems for both degenerate and nondegenerate systems. In almost all cases, the models can be used to determine the quantization of energy and eigenvalues for integrals associated with separation of variables in the original system. 2011 Article Models of Quadratic Algebras Generated by Superintegrable Systems in 2D / S. Post // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E70; 81R05; 17B80 DOI:10.3842/SIGMA.2011.036 http://dspace.nbuv.gov.ua/handle/123456789/146853 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this paper, we consider operator realizations of quadratic algebras generated by second-order superintegrable systems in 2D. At least one such realization is given for each set of Stäckel equivalent systems for both degenerate and nondegenerate systems. In almost all cases, the models can be used to determine the quantization of energy and eigenvalues for integrals associated with separation of variables in the original system.
format	Article
author	Post, S.
spellingShingle	Post, S. Models of Quadratic Algebras Generated by Superintegrable Systems in 2D Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Post, S.
author_sort	Post, S.
title	Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
title_short	Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
title_full	Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
title_fullStr	Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
title_full_unstemmed	Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
title_sort	models of quadratic algebras generated by superintegrable systems in 2d
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/146853
citation_txt	Models of Quadratic Algebras Generated by Superintegrable Systems in 2D / S. Post // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 27 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT posts modelsofquadraticalgebrasgeneratedbysuperintegrablesystemsin2d
first_indexed	2025-07-11T00:46:50Z
last_indexed	2025-07-11T00:46:50Z
_version_	1837309435617214464
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 036, 20 pages Models of Quadratic Algebras Generated by Superintegrable Systems in 2D? Sarah POST Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128 succ. Centre-Ville, Montréal (QC) H3C 3J7, Canada E-mail: post@crm.umontreal URL: http://crm.umontreal.ca/~post/ Received February 01, 2011, in final form March 24, 2011; Published online April 05, 2011 doi:10.3842/SIGMA.2011.036 Abstract. In this paper, we consider operator realizations of quadratic algebras generated by second-order superintegrable systems in 2D. At least one such realization is given for each set of Stäckel equivalent systems for both degenerate and nondegenerate systems. In almost all cases, the models can be used to determine the quantization of energy and eigenvalues for integrals associated with separation of variables in the original system. Key words: quadratic algebras; superintegrability; special functions; representation theory 2010 Mathematics Subject Classification: 22E70; 81R05; 17B80 1 Introduction In his 1968 book Lie theory and special functions, W. Miller Jr. used function space realizations of Lie algebras to establish a fundamental relationship between Lie groups and certain special functions including Bessel functions, hypergeometric functions and confluent hypergeometric functions [1]. It was further shown how the algebra relations can be used to identify special function identities. In this paper, we will apply these methods to the study of the representation theory for quadratic algebras generated by second-order superintegrable systems in 2D and their associated special functions. We would like to consider irreducible function space representations and so we restrict the Hamiltonian to a constant times the identity and construct difference or differential operator realizations for the elements of the algebra. We call such operator realizations models. A classical or quantum Hamiltonian on an m dimensional Riemannian manifold, with met- ric gjk, given respectively by H = 1 2 gjkpjpk + V (x1, x2), H = −~2 2 √ g ∂xj √ ggjk∂xk + V (x1, x2) is called superintegrable if it admits 2m−1 integrals of motion. For classical systems, we require that the integrals be functionally independent and, for quantum systems, we require that that they be algebraically independent within a Jordan algebra generated by xj , ∂xj and the identity. If both of the integrals are polynomial in the momenta in the classical case and as differential operators in the quantum case, we call the system nth-order superintegrable, where n is the maximal order of a minimal generating set of integrals. The study of superintegrability was pioneered by Smorodinsky, Winternitz and collaborators in the study of multiseparable systems on real Euclidean space [2, 3]. They identified all four ?This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html mailto:post@crm.umontreal http://crm.umontreal.ca/~post/ http://dx.doi.org/10.3842/SIGMA.2011.036 http://www.emis.de/journals/SIGMA/S4.html 2 S. Post mulitseparable potentials on real Euclidean space, since named the Smorodinsky–Winternitz po- tentials. Later, W Miller Jr. with collaborators including E. Kalnins, J. Kress, and G. Pogosyan published a series of papers which classified all second-order superintegrable systems in 2D (see e.g. [4, 5, 6] and references therein). In these paper, the authors proved that classical and quantum second-order superintegrable systems are in one to one correspondence and that the potentials are invariant with respect to scaling so that it is possible to normalize the −~2/2 term to be 1. Thus, for the remainder of the paper we will only consider quantum systems, except to make slight observations where the classical systems differ, and chose this normalization. It was further shown that any second-order superintegrable system in 2D can be related, via the Stäckel transform, to a superintegrable system on a space of constant curvature and a complete list was given of all second-order superintegrable systems on 2D Euclidean space, E2,C, and on the two sphere, S2,C. Since we will not use the Stäckel transform explicitly in this paper, we refer the reader to [7, 8] and references therein for a complete exposition. We only note here that the Stäckel transform is a mapping between Hamiltonian systems, possibly on different manifolds, which preserves superintegrability and the algebra structure of the integrals up to a permutation of the parameters and the energy. Thus, we can classify superintegrable systems based on the structure of their symmetry algebra. Such classifications have been worked out directly in [9, 10] and via the Stäckel transform [11, 12]. Most importantly for this paper, it was proved that the algebra generated by the constants of the motion for a second-order superintegrable system closes to form a quadratic algebra. That is, suppose H is second-order superintegrable with second-order integrals of the motion L1 and L2. The integrals L1 and L2 will not commute and we denote their commutator by R. The algebra to be considered is then the associative algebra generated by A = {L1, L2, H,R ≡ [L1, L2]} . Such an algebra is called a quadratic algebra if the commutator of any two elements can be written as at most a quadratic polynomial in the generators. Further, since the four generators can not be independent in the Jordan algebra generated by the operators xj and ∂xj and the identity, there will be a polynomial relation between them. The structure relations for our algebra are hence given by [L1, R] = P (L1, L2, H), [L2, R] = Q(L1, L2, H), R2 = S(L1, L2, H), (1.1) where P and Q are at most quadratic polynomials and S is at most cubic. We note that in the classical case, we will instead have a quadratic Poisson algebra and the highest order terms of the structure relations will be the same as in to the quantum case. If the system admits a first-order integral X, we call the system degenerate. In this case, the potential depends only on a single parameter and R can be expressed in terms of the basis X, L1, L2 and H [13]. The study of the the algebras generated by superintegrable systems and in particular their representation theory has been a subject of recent study (see e.g. [14, 15, 16, 17, 18]). The main advantage of the method of function space realization is that it can be used to find the eigenvalues of operators other than the Hamiltonian and to compute inter-basis expansion coefficients for the wave functions of the Hamiltonian. Further, as in the case of Lie algebras, the representation theory for quadratic algebras, and polynomial algebras more generally, seems to be intimately connected with special functions and their identities. This paper is divided up into four sections. In Section 2, we give a model which realizes the quadratic algebra associated to the singular isotropic oscillator, E1 or Smorodinsky–Winter- nitz i. A function space representation is given including the normalization and the weight function as well as the eigenfunctions and eigenvalues for each of the operators associated with separation of variables in the original system. We describe how the energy values for the system Models of Quadratic Algebras Generated by Superintegrable Systems in 2D 3 and eigenvalues of the operators associated to separation of variables are quantized in a finite dimensional representation. In Sections 3 and 4, we give at least one model for the algebras associated with a representative of each Stäckel equivalence class of nondegenerate and degen- erate systems respectively. In these sections, the model is given and we identify possible finite dimensional representations including those are associated with quantized values of the energy. In Section 5 there is a brief conclusion with possible future developments. 2 Exposition of a model: the singular isotropic oscillator In this section, we shall consider the quadratic algebra associated with the system E1/Smoro- dinsky–Winternitz i and construct an irreducible function space representation for algebra. The Hamiltonian for this system, the singular isotropic oscillator, is H = ∂2x + ∂2y − ω2 ( x2 + y2 ) + 1 4 − a 2 x2 + 1 4 − b 2 y2 . The remaining generators of the symmetry algebra are, L1 = ∂2x + 1/4− a2 x2 − ω2x2, L2 = (x∂y − y∂x)2 + (1/4− a2)y2 x2 + (1/4− b2)x2 y2 . The algebra relations are, recall R ≡ [L1, L2] , [R,L1] = 8L2 1 − 8HL1 + 16ω2L2 − 8ω2, (2.1) [R,L2] = 8HL2 − 8{L1, L2}+ ( 12− 16a2 ) H + ( 16a2 + 16b2 − 24 ) L1, (2.2) R2 = 8H{L1, L2} − 8 3 {L1, L1, L2}+ 16ω2L2 2 + 16 ( a2 − 1 ) H2 + ( 16a2 + 16b2 − 200 3 ) L2 1 − ( 32a2 − 200 3 ) HL1 − 176ω2 3 L2 − 4ω2 3 ( 48a2b2 − 48a2 − 48b2 + 29 ) . (2.3) In order for the representation to be irreducible, the Hamiltonian will be restricted to a con- stant and so, in the model, the action of the Hamiltonian is represented by a constant E, i.e. H ≡ E. Now, suppose that L1 is diagonalized by monomials, that is, it is of the form L1 = l1t∂t+ l0 where l1 and l0 are some constants to be determined later. Suppose also that the action of L2 on the basis of monomials is in the form of a three-term recurrence relation. From the algebra relations, it is straightforward to see that L2 cannot be modeled by a first-order operator or else the relations would be a Lie algebra. Thus, we take the Ansatz of a second-order differential operator with polynomial coefficients. The algebra relations (2.1)–(2.3) are realized by the following operators, H = E and L1 = −4ωt∂t + E − 2ω(1 + b), L2 = 1 2 t(8t+ 1)2∂2t − ( 16(E − 2ωb− 6ω)t2 + 2(E − 4ωb− 8ω) ω − 1 + b 2 ) ∂t + 2t((E − 4ω − 2ωb)2 − 4ω2a2) ω2 − 2E(1 + b) ω + 4b2 + 8b+ 5. As was proposed in the Ansatz, L1 is diagonalized by monomials in t and further the opera- tors L1 and L2 act as automorphisms on polynomials in t. Thus, we take the set {tk \| k = 0, . . .} to be the basis for our representation. The three-term recurrence relation generated by the action of L2 on monomials is L2t n = Cn+1,nt n+1 + Cn,nt n + Cn−1,nt n−1, (2.4) 4 S. Post where Cn+1,n = 2(E − 4ω − 2aω − 2bω − 4nω)(E − 4ω + 2aω − 2bω − 4nω) ω2 , Cn,n = −16n2ω + (4E − 16ω(1 + b))n− 2E(1 + b) + ω(4b2 + 8b+ 5) 2ω , Cn−1,n = n(n+ b) 2 . Note that for arbitrary parameters, C−1,0 vanishes whereas Cn+1,n in general does not vanish for any value of n and so the so the representation is infinite dimensional, bounded below. If we make the assumption that Cn+1,n is zero for some finite integer, then we obtain quantization conditions for the parameters and a finite representation of dimension m. In particular, we chose to solve Cm,m−1 = 0 for the energy value E to obtain E = −2ω(2m+ a+ b), m ∈ N. (2.5) The potential and hence the algebra relations are symmetric under the transformation b→ −b and a→ −a and so we could have taken the opposite sign for either a or b for the construction of the model and hence in the resulting quantization of the energy values. The existence of a three-term recursion formula as in (2.4) indicates the existence of raising and lowering operators. Indeed, they are given by the following relations A ≡ L2 + R 4ω − L2 1 2ω2 + E 2ω2 L1 − 1 2 = t∂2t + (1 + b)∂t, A† ≡ L2 − R 4ω − L2 1 2ω2 + E 2ω2 L1 − 1 2 = 64t3∂2t − 32(E − 2ωb− 6ω) ω t2∂t + 4((E − 2ωb− 4ω)2 − 4a2ω) ω2 t. Here we note that if L1 and L2 are self-adjoint, A and A† will only be mutual adjoints if ω is real, which is reflective of the physical fact that the potential will be attracting in that case. The commutation relations of the raising and lowering operators can either be determined from the quadratic algebra (2.1)–(2.3) or directly from the model. They are [L1, A] = 4ωA, [L1, A †] = −4ωA†, [A,A†] = − 4 ω3 L3 1 + 6E ω3 L2 1 − 2 ω3 ( E2 − 4ω2 ( a2 + b2 + 2 )) L1 − 8E(a2 − 1) ω . We return to finite dimensional model where the basis of eigenvectors for L1, is given by {φn(t) = knt n \|n = 0, . . . ,m− 1}. Let us assume the existence of an inner product for which L1 and L2 are self-adjoint, or practically, we assume A and A† are mutual adjoints. As mentioned above, this is equivalent to L1 and L2 being self-adjoint and the constants a, b, and ω being real. With this inner product, we can find the normalization for our eigenvectors using the raising and lowering operators. That is, we assume 〈Atn, tn−1〉 = 〈tn, A†tn−1〉 and 〈φn, φn〉 = 1 to obtain the recursion relation as k2n = 64(m− n)(m− n+ a) n(b+ n) k2n−1, Models of Quadratic Algebras Generated by Superintegrable Systems in 2D 5 so that kn = 16n √ (−m)n(−m− a)n n!(b)n which is real so long as a, b > 0. From these normalization coefficients, we can find a reproducing kernel for this Hilbert space which lies in the Hilbert space. It has the characteristic 〈δ(ts), f(t)〉 = f(s) and is given by ∑ φn(t)φn(s) = ∑ k2n(ts)n which is exactly the hypergeometric polynomial δ(t, s) = 2F1 ( −m, −m− a b ∣∣∣∣∣ ts ) . Finally, it is possible to construct an explicit function space representation for the inner product which will make L1 and L2 self-adjoint. We assume the inner product is of the form 〈f(t), g(t)〉 = ∫ γ f(t)g(t)ρ(t, t)dtdt where γ is a path to be determined later. We can determine the weight function using the following relation, 〈L1f, g〉 = ∫ γ (L1f(t))g(t)ρ(t, t)dtdt = ∫ γ f(t)(L1g(t))ρ(t, t)dtdt = 〈f, L1g〉. Similarly, we require 〈Af, g〉 = 〈f,A†g〉. These two integral equations can be transformed by integration by parts into differential equation for the weight function whose solutions are ρ(tt) = c1 2F1 ( 1 +m, m+ 1 + a 1− b ∣∣∣∣∣ tt ) + c2(tt) b 2F1 ( 1 +m+ b, m+ 1 + a+ b 1 + b ∣∣∣∣∣tt ) . Finally, we note that we can also diagonalize the operator L2 in the model. The solutions of the eigenvalue equation for L2, (L2 − λ)ψλ = 0, are hypergeometric functions, 2F1’s [19]. If we assume the model is finite dimension and restrict E to the values in (2.5), then L2 becomes L2 = t(8t− 1)2 2 ∂2t − (8t+ 1)((2m+ a− 3)8t− b− 1) 2 ∂t + 32(m− 1)(m− 1 + a)t− 4m(b+ 1)− 2ba− 2a+ 2b+ 5 2 . For a finite dimensional irreducible representation, the hypergeometric series must terminate and we obtain a quantization relation on the eigenvalues λ = −3/2−2b−2a−4k−2ba−4bk−4ak−4k2 and the basis functions become, for k = 1, . . . ,m− 1, ψk(t) = lk(8t+ 1)m−1−k 2F1 ( −k, −a− k 1 + b ∣∣∣∣∣− 8t ) = lk(8t+ 1)m−1P b,ak ( 1− 8t 1 + 8t ) . Here the lk’s are normalization constants and P b,ak is a Jacobi polynomial. This model is an interesting example of the simplicity of differential models. We can di- rectly compute the eigenvalues for all the operators and the normalizations of the basis vectors. Also, the kernel function and integral representation of the inner product give useful tools to find expansion formulas and possible generating functions for the original quantum mechanical problem. In the remainder of the paper, we exhibit a function space realization for a representative of each of the Stäckel equivalence classes of second-order superintegrable systems in 2D. 6 S. Post 3 Models of quadratic algebras for non-degnerate second-order superintegrable systems We will make use of the following conventions. On Euclidean space, the generators of the Killing vectors are, px = ∂x, py = ∂y, M = x∂y − y∂x. We also define the operator p± ≡ ∂x ± i∂y. The Laplacian on E2,C in Cartesian coordinates is ∆ ≡ ∂2x + ∂2y . In complex coordinates, z = x+ iy, z = x− iy, we have ∆ ≡ ∂z∂z. Or, we can take a real form in lightlike coordinates, ν = x, ζ = iy, to obtain the wave operator, ∆ ≡ ∂2ν − ∂2ζ . For systems on the two sphere, we use the coordinates of the standard embedding of the sphere into 3 dimensional Euclidean space. We denote these s1, s2, s3 such that s21 +s22 +s23 = 1. The basis for the Killing vectors is Ji = ∑ i,j,k εijksj∂sk . The Laplacian on S2,C is ∆S2 ≡ 3∑ i=1 J2 i . The quantum algebra structures often have symmetrized terms. We define these by {a, b} ≡ ab+ba and {a, b, c} ≡ abc+acb+bac+bca+cab+cba. Also used is the permutation sign function εijk and εijkl, the completely skew-symmetric tensor on three or four variables respectively. In the following sections, we exhibit irreducible representations of the quadratic algebras, defined by relations (1.1), using both differential and difference operator realizations of the operators acting on function spaces. Because the Hamiltonian operator commutes with all of operators, it must be a constant for any irreducible representation and so we restrict the Hamiltonian to a constant energy H ≡ E. We focus on models which will diagonalize operators associated with separation of variables, since these are of immediate interest for explicit solution of the physical system though additional models are given as well. Most of the models were obtained by assuming a differential operator Ansatz except in the case of S9 which is realized as a difference operator. In this case, the operators were derived from the abstract structural relations though it is interesting to note that they also could have been obtained through the quantization of a model for the Poisson algebra for the associated classical system [20, 21]. Unless otherwise noted, these models were first exhibited in [22]. We classify the nondegenerate systems in 2D by their Stäckel equivalence classes and identify them by the leading order terms of the functional relation, as seen in the accompanying Table 1. This classification comes from [11] and the nomenclature of the Ei’s comes from [4]. In this section, we describe the symmetry operators, quadratic algebras and at least one model for each of the equivalence classes. Models of Quadratic Algebras Generated by Superintegrable Systems in 2D 7 Table 1. Stäckel equivalence classes of non-degenerate systems in 2 dimensions. Leading terms of Casimir relation System Operator models L3 1 + f(αi, H)L2 2 E2, S1 Differential L3 1 + f(αi, H)L1L2 E9, E10 Differential L3 1 + 0 E15 Differential L2 1L2 + f(αi, H)L2 2 E1, E16, S2, S4 Differential L2 1L2 + 0 E7, E8, E17, E19 Differential L1L2(L1 + L2) + f(αi, H)L1L2 S7, S8, S9 Difference 0 + f(αi, H)L1L2 E3, E11, E20 Differential 3.1 E2: Smorodinsky–Winternitz ii The Hamiltonian is on real Euclidean space for real constants ω, b, c H = ∆− ω2(4x2 + y2) + bx+ 1 4 − c 2 y . A basis for the symmetry operators is given by H and L1 = p2x − 4ω2x2 + bx, L2 = 1 2 {M,py} − y2 ( b 4 − xω2 ) + ( 1 4 − c2 ) x y2 . The symmetry algebra relations are, recall R ≡ [L1, L2], [L1, R] = −2bH + 16ω2L2 + 2bL1, (3.1) [L2, R] = 8L1H − 6L2 1 − 2H2 + 2bL2 − 8ω2 ( 1− c2 ) , (3.2) R2 = 4L3 1 + 4L1H 2 − 8L2 1H + 16ω2L2 2 − 4bL2H + 2b{L1, L2} + 16ω2 ( 3− c2 ) L1 − 32ω2H − b2 ( 1− c2 ) . (3.3) The algebra relations (3.1)–(3.3) are realized by H = E and the following operators, L1 = 4tω∂t + 2ω + 1 16 b2 ω2 , L2 = 32ωt3∂2t + (( 16(E − 6ω) + b2 ω2 ) t2 − b 2ω t− 1 8 ) ∂t + (16Eω2 − 64ω3 − b2)2 − (32ω3c)2 128ω5 t+ 16Eω2 − 32ω3 − b2 128ω4 . Here, the eigenfunctions of L1 are monomials and the action of L2 on this basis can be represented as a three-term recurrence relation given by L2t n = Cn+1,nt n+1 + Cn,nt n + Cn−1,nt n−1, where Cn+1,n = (16Eω2 − 64ω3 − b2 − 64ω3n)2 − (32ω3c)2 16ω2 , Cn−1,n = n 8 , Cn,n = b(16Eω2 − 32ω3 − b2 − 64ω2n) 128ω4 . Notice, that C−1,0 vanishes and so our representation is bounded below. Further, if there exists some n so that Cn+1,n also vanishes then the representation becomes finite dimensional. 8 S. Post Conversely, we can assume that the representation is finite dimensional to obtain quantization conditions on the energy. If we assume that representation space is m dimensional and spanned by the monomials {tn \|n = 0, . . . ,m−1}, then the restriction Cm,m−1 = 0 gives the quantization of the energy values E = 4ω(m+ 2εc) + b2 16ω2 , m ∈ N. Here, we can take either ε = ±1 in the energy which is consistent with the Hamiltonian depending only c2. Finally, we note that we can define raising and lowering operators in this model as, A = L2 − R 4ω + bL1 4ω2 − bE 4ω2 = 1 ω2 ∂t, A† = L2 + R 4ω + bL1 4ω2 − bE 4ω2 = t ( 64ωt2∂3t − 128ω3(16Eω2 − 96ω3 − b2) 64ω5 t∂t + ( 16Eω2 − 64ω3 − b2 )2 − (32ω3c )2) . The raising and lowering operators obey the following commutation relations [L1, A] = −4ωA, [L1, A †] = 4ωA†, [A,A†] = − 3 ω L2 1 + 32ω2 + b 8ω3 L1 − 1 8ω3 ( 8E2ω2 + b2E + 32ω4 ( 1− c2 )) . 3.2 E10 The Hamiltonian for this system is given by, with z = x+ iy, z = x− iy H = ∆ + αz + β ( z − 3 2 z2 ) + γ ( zz − 1 2 z3 ) . A basis for its symmetry operators is given by H and L1 = p2− + γz2 + 2βz, L2 = 2i{M,p−}+ p2+ − 4βzz − γzz2 − 2βz3 − 3 4 γz4 + γz2 + αz2 + 2αz. The algebra relations are given by [R,L1] = −32γL1 − 32β2, [R,L2] = 96L2 1 − 128αL1 + 32γL2 + 64βH + 32α2, R2 = 64L3 1 + 32γ{L1, L2} − 128αL2 1 − 64γH2 − 128βHL1 + 64β2L2 + 64α2L1 − 128βαH − 256γ2. This algebra can be transformed into the Lie algebra sl2 by using the following invertible trans- formation K1 = L1 + β2 γ , K2 = L2 + 1 γ L2 1 − β2 + 2αγ γ2 L1 + 2β γ H + (αγ + β2)2 γ3 . In this basis, the algebra relations reduce to a Lie algebra [R,K1] = −32γK1, [R,K2] = 32γK2, [K1,K2] = R, (3.4) R2 = 32γ{K1,K2} − 64γH2 − 128β(αγ + β2) γ H − 64(β6 + 4γ4 + α2β2γ2 + 2αβ4γ) γ3 . (3.5) Models of Quadratic Algebras Generated by Superintegrable Systems in 2D 9 The algebra relations (3.4), (3.5) are realized by H = E and the following operators, K1 = 16γ∂t, K2 = t2∂t + ( 1 + √ −γ(γ2E + αβγ + β3) 2γ3 ) t, R ≡ [L1, L2] = 16γ ( 2t∂t + ( 1 + √ −γ(γ2E + αβγ + β3) 2γ3 )) . The operator R is diagonalized by monomials tn and in this basis K1 and K2 act as lowering and raising operators respectively. Note that K1 annihilates constants and so the representation is bounded below. On the other hand, for a finite dimensional representation, we require that there exist some integer m such that K2t m−1 = ( m− 1 + ( 1 + √ −γ(γ2E + αβγ + β3) 2γ3 )) tm = 0. This leads to the quantization condition on the energy E = 2i √ γm− αβ γ − β3 γ , m ∈ N. (3.6) Notice that in order to obtain a real energy value we require that all the parameters be real and that γ < 0. It is also possible to diagonalize a linear combination of K1 and K2. For example, the solutions of the eigenfunction equation (K1 +K2 − λ)Ψ = 0 are Ψ = ( 16γ + t2 )√−γ(γ2E+αβγ+β3) 4γ3 exp ( λ 4 √ γ arctan ( t 4 √ γ )) which, for finite dimensional representations where E is restricted to the value in (3.6), give quantization conditions on the eigenvalues λ. That is, if we require that Ψ be a polynomial in t of degree less that m we obtain a complete set of eigenfunctions given by (K1 +K2)Ψn = λnΨn, n = 0, . . . ,m− 1, Ψn = ( 4 √ −γ + t )n( 4 √ −γ − t )m−n−1 , λn = 4 √ −γ(m− 2n− 1). Finally, we can return to the original basis of L1 and L2. In the model, L1 = 16γ∂t − β2 γ , L2 = 256γ∂2t + ( t2 + 32α+ 48β2 γ ) ∂t + ( 1 + √ γ(γ2E + αβγ + β3) 2γ3 ) t− 2βγ2E + α2γ2 + 4αβ2γ + 3β4 γ3 . 3.3 E15 Here H = ∆ + h(z), 10 S. Post where the potential is an arbitrary function of z. A basis for the symmetry operators is L1 = p2−, L2 = 2i{M,p−}+ i ∫ z dh dz dz. The only nonzero algebra relation is [L1, L2] = iL1. This system is unique among all 2 dimen- sional superintegrable systems in that the symmetry operators are not functionally linearly in- dependent and do not correspond to multiseparability. The only separable system is determined by diagonalizing L1, essentially z, z, and this coordinate system is not orthogonal. A model is L1 = d dt + a, L2 = it d dt + iat, but the irreducible representations of the algebra yield no spectral information about H. Again, since this algebra is a Lie algebra, the above model is derivative of one found in [1]. 3.4 E1, Smorodinsky–Winternitz i This is the system considered in Section 2 and for completeness we recall the results here. The Hamiltonian for the system is H = ∆− ω2 ( x2 + y2 ) + 1 4 − a 2 x2 + 1 4 − b 2 y2 . The remaining generators of the symmetry algebra are L1 = ∂2x + 1/4− a2 x2 − ω2x2, L2 = M2 + (1/4− a2)y2 x2 + (1/4− b2)x2 y2 . The algebra relations are [R,L1] = 8L2 1 − 8HL1 + 16ω2L2 − 8ω2, (3.7) [R,L2] = 8HL2 − 8{L1, L2}+ ( 12− 16a2 ) H + ( 16a2 + 16b2 − 24 ) L1, (3.8) R2 = 8H{L1, L2} − 8 3 {L1, L1, L2}+ 16ω2L2 2 + 16 ( a2 − 1 ) H2 + ( 16a2 + 16b2 − 200 3 ) L2 1 − ( 32a2 − 200 3 ) HL1 − 176ω2 3 L2 − 4ω2 3 ( 48a2b2 − 48a2 − 48b2 + 29 ) . (3.9) The algebra relations (3.7)–(3.9) are realized by H = E, and the following operators L1 = −4ωt∂t + E − 2ω(1 + b), L2 = 1 2 t(8t+ 1)2∂2t − ( 16(E − 2ωb− 6ω)t2 + 2(E − 4ωb− 8ω) ω − 1 + b 3 ) ∂t + 2t((E − 4ω − 2ωb)2 − 4ω2a2) ω2 − 2E(1 + b) ω + 4b2 + 8b+ 5. As described in the previous section, the eigenvalues for L1 are monomial and the eigen- functions for L2 are hypergeometric functions which reduce to Jacobi polynomials for finite representations. The representation become finite dimensional under the quantization of energy E = −2ω(2m+ a+ b), (2.5). Models of Quadratic Algebras Generated by Superintegrable Systems in 2D 11 3.5 E8 The Hamiltonian is H = ∆ + αz z3 + β z2 + γzz. A basis for the symmetry operators is given by H and L1 = p2− + z4 − α z2 , L2 = M2 + β z z + α z2 − z2 z2 . The algebra relations are [R,L1] = 8L1 2 + 32αγ, [R,L2] = −8{L1, L2}+ 8bH − 16(α+ 1)L1, (3.10) R2 = −8 3 {L1, L1, L2} − ( 16α+ 176 3 ) L2 1 + 16αH2 − 64αγL2 + 16βL1H − 64γα2 − 16γβ2 + 64 3 αγ. (3.11) The algebra relations (3.10), (3.11) are realized by H = E and the following operators L1 = 2 √ −αγt, L2 = −4 ( t2 − 1 ) ∂2t + (( 2β√ α − 8 ) t+ 2E√ −γ ) ∂t − ( 1 + β 2 √ α )2 − α. Here, the eigenfunctions of L2 are hypergeometric functions which restrict to Jacobi polynomials under quantization of eigenvalues L2Ψn = λnΨn, λn = −4n2 + (8β √ α− 16α) 4α n− 4α2 + 4α+ 4β √ α− β2 4α . The eigenfunctions are given by Ψn = lnP a,b n (−t), a = E 4 √ −γ − β 4 √ α , b = − E 4 √ −γ − β 4 √ α , where ln is a normalization constant. The action of L1 in this model is via multiplication by the variable t and gives a three-term recurrence formula. Note that for some quantization of the energy the eigenfunctions become singular for all n ≥ m. This occurs under the quantization condition m ∈ N with energy eigenvalues E = 2 √ −γ ( 2m+ 2± β 2 √ α ) , m ∈ N. 3.6 S9: the generic system in 2D The Hamiltonian is H = ∆S2 + 1 4 − a 2 s21 + 1 4 − b 2 s22 + 1 4 − c 2 s23 . A basis for the symmetry operators is L1 = J2 3 + ( 1 4 − a2 ) s21 s22 + ( 1 4 − c2 ) s22 s21 , L2 = J2 1 + ( 1 4 − a2 ) s23 s22 + ( 1 4 − b2 ) s22 s23 , H = L1 + L2 + L3 + 3 4 − a2 − b2 − c2. 12 S. Post The structure equations can be put in the symmetric form using the following identifications a1 = 1 4 − c2, a2 = 1 4 − a2, a3 = 1 4 − b2, [Li, R] = εijk (4{Li, Lk} − 4{Li, Lj} − (8 + 16aj)Lj + (8 + 16ak)Lk + 8(aj − ak)) , (3.12) R2 = 8 3 {L1, L2, L3} − (16a1 + 12)L2 1 − (16a2 + 12)L2 2 − (16a3 + 12)L2 3 + 52 3 ({L1, L2}+ {L2, L3}+ {L3, L1}) + 1 3 (16 + 176a1)L1 + 1 3 (16 + 176a2)L2 + 1 3 (16 + 176a3)L3 + 32 3 (a1 + a2 + a3) + 48(a1a2 + a2a3 + a3a1) + 64a1a2a3. (3.13) We can obtain L1 in the model by using the following difference operator, based upon the Wilson polynomial algebra. This model is unique in that the energy values E and eigenvalues of the operators were used in determining the model. We define the coefficients α, β, γ and δ as α = −a+ c+ 1 2 − µ, β = a+ c+ 1 2 , γ = a− c+ 1 2 , δ = a+ c− 1 2 + b+ µ+ 2, and use difference operators TAF (t) = F (t+A), τ = 1 2t (T 1/2 − T−1/2), τ∗ = 1 2t [ (α+ t)(β + t)(γ + t)(δ + t)T 1/2 − (α− t)(β − t)(γ − t)(δ − t)T−1/2 ] . The algebra relations (3.12), (3.13) are realized by the following operators L3 = −4t2 + a2 + c2, L1 = −4τ∗τ − 2(a+ 1)(b+ 1) + 1 2 , H = E, E ≡ −1 4 (4µ+ 2a+ 2b+ 2c+ 5)(4µ+ 2a+ 2b+ 2c+ 3) + 3 2 − a2 − b2 − c2. The model realizes the algebra relations for arbitrary complex µ and restricts to a finite dimensional irreducible representation when µ = m ∈ N. In this model, we obtain spectral resolution of L3 with delta functions as eigenfunctions. The eigenfunctions of L1 are Racah polynomials in the finite dimensional case and Wilson polynomials for the infinite dimensional, bounded below case. This model was first published in [23] where the model was worked out in full generality including the normalizations and the weight functions. It has also been recently been extended to the 3D analog in [16]. 3.7 E20: Smorodinksy–Winternitz iii The Hamiltonian is H = ∆ + 4√ x2 + y2 α+ β √√ x2 + y2 + x√ x2 + y2 + γ √√ x2 + y2 − x√ x2 + y2  . In parabolic coordinates (x′, y′), with x′ = √ x2 + y2 + x, y′ = √ x2 + y2 − x, the Hamiltonian can be written as H = 1 x′2 + y′2 ( ∂2x′ + ∂2y′ ) + 4(α− βx′ − γy′) x′2 + y′2 , Models of Quadratic Algebras Generated by Superintegrable Systems in 2D 13 The integrals are L1 = 1 x′2 + y′2 ( y′2∂2x′ − x′2∂2y′ − 2α ( x′2 − y′2 ) − 4βx′y′2 + 4γx′2y′ ) , L2 = 1 x′2 + y′2 ( −x′y′ ( ∂2x′ + ∂2y′ ) + ( x′2 + y′2 ) ∂x′∂y′ − 4αx′y′ + 2 ( x′2 − y′2 ) (y′β − x′γ) ) . The algebra relations are given by, [R,L1] = −4L2H + 16βγ, [R,L2] = 4L1H − 8 ( β2 − γ2 ) , (3.14) R2 = 4L2 1H + 4L2 2H + 4H2 − 16α2H + 16 ( γ2 − β2 ) L1 − 32βγL2 − 32α2 ( β2 + γ2 ) . (3.15) Notice, these restrict to a Lie algebra on a constant energy surface, H = E, and so the model described below is not new but, for example, can be derived from those given in [1]. The algebra relations (3.14), (3.15) are realized by H = E and the following operators, L1 = −2 √ Et∂t − √ E + 2α+ 4 β2 E , L2 = −1 2 Et2∂t + 2∂t − 1 2 tE + √ Eat+ ( β2√ E + γ2√ E ) t− 4 βγ√ E . Here L1 is diagonalized by monomials tn and the action of L2 on monomials is given by the following three-term recursion formula L2t n = Cn+1,nt n+1 + Cn,nt n + Cn−1,nt n−1 with Cn+1,n = −(n+ 1)E + α√ E + β2 + γ2√ E , Cn,n = 4βγ E , Cn−1,n = 2n. Notice, that C−1,0 vanishes for arbitrary parameters and so our representation is bounded below. On the other hand, if we require that the representation be finite dimensional, say of dimension m, we obtain the restriction −mE + 2 √ Ea+ β2√ E + γ2√ E = 0, m ∈ N, (3.16) which gives the quantized restrictions on the energy. The eigenfunctions for L2 are given by Ψ = ( Et2 − 4 )E− 3 2 (Eα+β2+γ)− 1 2 exp ( λE − 4βγ E 3 2 arctanh (√ Et 2 )) , where L2Ψ = λΨ. If we assume that the model is finite dimensional, i.e. that the energy value E satisfies (3.16), then the eigenvalues are quantized as λ = λn ≡ 4βγ E + (m− 1− 2n) √ E and there exists a complete set of eigenfunctions Ψn satisfying L2Ψn = λnΨn, Ψn = (√ Et− 2 )m−n−1(√ Et− 2 )n . 14 S. Post This model admits raising and lowering operators, A = L2 + R 2 √ E − βγ E = 4∂t, A† = L2 − R 2 √ E − βγ E = −Et2∂t + 2αE − E 3 2 + 2β2 + 2γ2√ E t, which satisfy the commutation relations [A,L1] = −2 √ EA, [A†, L1] = 2 √ EA†, [A,A†] = −2L2 1 + β2 − γ2 E L1 − 2E + 8α2 + 16α(β2 + γ2) E + 32β2γ2 E2 . 4 2D degenerate systems Next, we consider degenerate systems whose symmetry algebra includes a first-order integral, X. As described above, this requires that the potential depend on only one parameter, not including the trivial additive constant. In these systems, the symmetry algebra is defined by the operators H, L1, L2 and X. It is always possible to rewrite the commutator R = [L1, L2] as a polynomial in the other operators of maximal degree 3 in X and 1 in L1, XL1, L2, XL2, H and XH. In these systems, the commutation relations are in terms of X instead of R. That is, the defining relations are [L1, X] = P1 ( L1, L1, X 2, X,H ) , [L1, X] = P2 ( L1, L1, X 2, X,H ) , [L1, L2] = Q ( X3, XL1, XL2, XH,L1, L2, H,X ) . Here, the Pi’s and Q are linear in the arguments. Furthermore, the functional relation is no longer in terms of R2 but instead a fourth-order identity. We begin with the table of the equivalence classes of degenerate systems; there are exactly 6 degenerate systems in 2 dimensions. Table 2. Stäckel equivalence classes of degenerate systems which admit a Killing vector X, in 2D. Leading order terms System Operator models 0 + L1L2 +AX2 E3, E18 Differential X4 + L1L2 S3, S6 Differential and Difference X4 +X2L1 + L2 2 + 0 E12, E14 Differential 0 +X2L1 + L2 2 +AL1 E6, S5 Differential X4 + L2 1 E5 Differential 0 +X2L1 + L2 E4, E13 Differential 4.1 E18: the Coulomb system in 2D This system is defined by the Hamiltonian H = ∆ + α√ x2 + y2 . A basis for the symmetry operators is formed by H, X = M and L1 = 1 2 {M,px} − αy 2 √ x2 + y2 , L2 = 1 2 {M,py} − αx 2 √ x2 + y2 . Models of Quadratic Algebras Generated by Superintegrable Systems in 2D 15 The symmetry algebra is defined by the following relations [L1, X] = L2, [L2, X] = −L1, [L1, L2] = HX, (4.1) L2 1 + L2 2 −HX2 + H − α2 4 = 0. (4.2) We can change basis so that the algebra is in the standard form of the Lie algebra sl2. With the substitutions A = L1 + iL2, A † = L1 − iL2 the algebra relations (4.1), (4.2) become, [A,X] = −iA, [A†, X] = iA†, [A,A†] = 2iHX, (4.3) {A,A†} − 2HX2 + H 2 − α2 2 = 0. (4.4) The Lie algebra defined by (4.3), (4.4) is realized by H = E and the following operators X = −i ( t∂t + 1 2 + α 2 √ E ) , A = E 4 ∂t, A† = −4t ( t∂t + 1 + α√ E ) . The operator X is diagonalized by monomials tn and for arbitrary E and α the representa- tion is infinite and bounded below. Finite dimensional representation occur if A† annihilates a monomial. That is A†tm−1 = −4(Em− α √ E) E tm = 0 when the energy value E takes the values E = α2 m2 , m ∈ N. Again, since this algebra is a Lie algebra the model employed is not new. 4.2 S3 This system is defined by the Hamiltonian H = ∆S2 + 1 4 − a 2 s23 . A basis for the symmetry operators is formed by H, X = J3 and L1 = J2 1 + (14 − a 2)s22 s23 , L2 = 1 2 (J1J2 + J2J1)− (14 − a 2)s1s2 s23 . The algebra relations are given by [L1, X] = 2L2, [L2, X] = −X2 − 2L1 +H + a2 − 1 4 , (4.5) [L1, L2] = −{L1, X}+ ( 2a2 − 1 ) X, (4.6) 1 6 {L1, X,X} −HL1 + L2 2 + L2 1 − ( a2 − 7 6 ) X2 − ( a2 + 5 12 ) L1 + H 6 − 5 24 ( 4a2 − 1 ) = 0. (4.7) The algebra relations (4.5)–(4.7) are realized by H = E and the following operators X = 2it∂t + ic0, 16 S. Post L1 = t(t+ 1)2∂2t + (t+ 1)(c1t+ c1 + 2c0 − 1)∂t + ( c21 + 3c20 2 − 2c0c1 + 3c0 − 3c1 − a2 2 + E 2 + 19 18 ) t+ a2 + c20 + E 2 − 1 8 , L2 = −i(t3 − t)∂2t − i ( c1t 2 + 2 + 2c0 − c1 ) ∂t − i ( c21 + 3c20 2 − 2c0c1 + 3c0 − 3c1 − a2 2 + E 2 + 19 18 ) t, where, for compactness of the equations, we have chosen constants c0, c1 as c0 = a− 1 + √ 4E − 1 2 , c1 = 3 4 ( 2a+ √ 4E − 1 ) + 1 4 √ 4a2 + 16a− 12E − 13− 4(a− 2) √ 4E − 1. In the model, X is diagonalized with monomials and both L1 and L2 have a three-term recurrence relation in this basis given by L1t n = Cn+1,nt n+1 + Cn,nt n + Cn−1,nt n−1, (4.8) L2t n = iCn+1,nt n+1 − iCn−1,ntn−1, (4.9) with Cn+1,n = n2 + (c1 − 1)n+ c21 + 3c20 2 − 2c0c1 − 3c1 + 3c0 − a2 + E 2 + 19 8 , Cn,n = 2n2 + 2nc0 + c20 + E + a2 2 − 1 8 , Cn−1,n = n(n− c1 + 2c0 + 1). Note that, for arbitrary parameters, C−1,0 vanishes and so the model is bounded below. On the other hand, if we require that the representation space be finite dimensional, i.e. Cm,m−1 = 0, we obtain the quantization condition E = −(m− a)2 + 1 4 , m ∈ N. (4.10) As can be seen directly form the three-term recurrence formulas (4.8) and (4.9), there exist raising and lowering operators given by A† = L1 + iL2 + 1 2 ( X2 − E + 1 4 − a2 ) , A = L1 − iL2 + 1 2 ( X2 − E + 1 4 − a2 ) . The symmetry algebra of the raising and lowering operators is given by [A,X] = 2iA, [A†, X] = −2iA†, {A,A†} − 1 2 X4 +X2 ( E − a2 + 11 4 ) + 1 32 ( 4E + 4a2 + 8a+ 3 )( E + 4a2 − 8a+ 3 ) = 0. There is also a difference operator model associated with the spectral resolution of L1. The operators are formed from the operators T k defined as T kf(t) = f(t + k). When restricted to finite dimensional representations, i.e. the energy E takes the values as in (4.10), the algebra relations (4.5)–(4.7) are realized by L1 = −t2 + a2 − 1 4 , −iX = (1/2− a− t)(m+ a− 1/2− t) 2t T 1 − (1/2− a+ t)(m+ a− 1/2 + t)) 2t T−1, Models of Quadratic Algebras Generated by Superintegrable Systems in 2D 17 L2 = (1− 2t)(1/2− a− t)(m+ a− 1/2− t) 4t T 1 + (1 + 2t)(1/2− a+ t)(m+ a− 1/2 + t)) 4t T−1. The eigenfunctions of L1 are delta functions and the eigenfunctions of L2 are dual Hahn polyno- mials. This model was first published in [20] where the model was worked out in full generality including the normalizations and the weight functions. 4.3 E14 The Hamiltonian is, with z = x+ iy, z = x− iy, H = ∆ + α z2 . A basis for the symmetry operators is formed by H, X = p− and L1 = 1 2 {M,p−}+ α iz , L2 = M2 + αz z , The symmetry algebra is [X,L1] = iX2, [X,L2] = 2iL2, [L1, L2] = i{X,L2}+ i 2 X, (4.11) L2 1 − 1 2 {L2, X 2}+ αH − 5 4 X2 = 0. (4.12) The algebra relations (4.11), (4.12) are realized by H = E and the following operators X = 1 t , L1 = i∂t, L2 = −t2∂2t − 2t∂t + αEt2 − 1 4 . Eigenfunctions for L2 are Bessel functions while eigenfunctions of L1 are exponentials both with continuous spectrum. This model has no obvious finite dimensional restrictions. 4.4 E6 The Hamiltonian is H = ∆ + 1 4 − a 2 x2 . A basis for the symmetry operators is formed by H, X = py and L1 = 1 2 {M,px} − (14 − a 2)y x2 , L2 = M2 + (14 − a 2)y2 x2 , The algebra relations are given by [L1, X] = H −X2, [L2, X] = 2L1, [L1, L2] = {X,L2}+ (1− 2a)X, (4.13) L2 1 −HL2 − 2{L1, X}+ H 2 + ( 1 2 − a2 ) X2 = 0. (4.14) The algebra relations (4.13), (4.14) are realized by H = E and the following operators X = √ E ( t2∂t + (a+ 1)t+ 1 ) , L1 = − √ E ( t3∂2t + ((2a+ 3)t+ 2)t∂t + (a+ 1)2t+ a+ 1 ) , 18 S. Post L2 = −t2∂2t − 2((a+ 1)t+ 1)∂t − a− 1 2 . Note that the operators X, L1 and L2 act on monomials tn via three-term recurrence relations Xtn = √ E(n+ a+ 1)tn+1 + √ Etn, L1t n = − √ E(n+ a+ 1)2tn+1 − √ E(2n+ a+ 1)tn, L2t n = − ( n2 + (n+ 2a) + a+ 1 2 ) tn − 2ntn−1. Thus, the model can be realized as a bounded below representation using the set of mono- mial {tn \|n ∈ N}. The representation will become finite dimensional, with dimension m if the parameter a is restricted to quantized values a = −m. Notice, a constant is quantized instead of the energy which does not make sense in the physical system, since the constants should be given by the system. However, under a Stäckel transform the constant and the energy will be interchanged so our model is giving quantization levels of the energy of a Stäckel equivalent system; in this case, S5. The eigenfunctions of L1 are gauge equivalent to Laguerre polynomials with eigenvalues √ E(2n−m+ 1). The eigenfunctions of L2 are gauge equivalent to generalized Laguerre polynomials with eigenvalues m2 − k2 + k − 1 2 . 4.5 E5 The Hamiltonian is H = ∆ + αx. A basis for the symmetry operators is formed by H, X = py and L1 = pxpy + 1 2 αy, L2 = 1 2 {M,py} − 1 4 αy2, The algebra relations are given by [L1, X] = −α 2 , [L2, X] = L1, [L1, L2] = 2X3 −HX, (4.15) X4 −HX2 + L2 1 + αL2 = 0. (4.16) The algebra relations (4.15), (4.16) are realized by H = E and the following operators, X = t, L1 = −α 2 ∂t, L2 = −1 4 α∂2t + 1 α t2 ( E − t2 ) . Here the only nontrivial eigenfunction is that of L2 which gives solutions of the triconfluent Heun equation. This model gives no obvious finite dimensional restrictions. 4.6 E4 The Hamiltonian is H = ∆ + α(x+ iy). A basis for the symmetry operators is formed by H, X = p+ and L1 = p2x + αx, L2 = 1 2 {M,p+}+ iα 4 (x+ iy)2, Models of Quadratic Algebras Generated by Superintegrable Systems in 2D 19 The algebra relations are given by [L1, X] = −α, [L2, X] = iX2, [L1, L2] = −iX3 − iHX + i {L1, X} , (4.17) X4 − 2 3 {L1, X,X}+ 2HX2 +H2 + 4αiL2 = 0. (4.18) The algebra relations (4.17), (4.18) are realized by H = E and the following operators X = iαt, L1 = i∂t + E 2 , L2 = −αt2∂t + iα4t4 − 4α2t+ iE2 4α . This model gives no obvious finite dimensional restrictions and the basis for the representation can be chosen as exponential functions. 5 Conclusion In this paper, we have demonstrated at least one function space realization for each class of Stäckel equivalence superintegrable systems. We have shown that these quadratic algebras are related to various families of special functions and orthogonal polynomials. There are several possible directions that this analysis can be extended and the model de- scribed above can be further applied and analyzed. One possible direction, is to use the fact that there exist appropriately chosen limit processes which takes superintegrable system S9 on the sphere to all other superintegrable systems in 2D (see e.g. [24, 25]). It would be interesting to see how these limits are manifested at the level of the algebras and particularly the models. Also of interest would be to consider the restrictions from nondegenerate to degenerate systems and to study the effect of such a limit on the models. These methods can also be applied to the construction and analysis of quadratic algebras generated by superintegrable systems in higher dimensions. In particular, the quadratic algebras for all real nondegenerate 3D superintegrable systems were shown to contain various copies of quadratic algebras from two dimensional systems [26]. As was the case in [16, 27], the models for the 2D system can be used as a basis for the models of the 3D system. It would be interesting to see how the algebra decomposed with respect to the subalgebras and whether that has any effect on the models obtained and possible special function relations. Acknowledgements S.P. acknowledges a postdoctoral fellowship awarded by the Laboratory of Mathematical Physics of the Centre de Recherches Mathématiques, Université de Montréal. References [1] Miller W. Jr., Lie theory and special functions, Mathematics in Science and Engineering, Vol. 43, Academic Press, New York – London, 1968. [2] Fris I., Mandrosov V., Smorodinsky Ya.A., Uhĺır M., Winternitz P., On higher symmetries in quantum mechanics, Phys. Lett. 16 (1965), 354–356. [3] Makarov A.A., Smorodinsky Ya.A., Valiev Kh., Winternitz P., A systematic search for non-relativistic system with dynamical symmetries, Nuovo Cim. A 52 (1967), 1061–1084. [4] Kalnins E.G., Pogosyan G.S., Miller W. Jr., Completeness of multiseparable superintegrability in two di- mensions, Phys. Atomic Nuclei 6 (2002), 1033–1035. [5] Kalnins E.G., Kress J.M., Miller W. Jr., Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory, J. Math. Phys. 46 (2005), 053509, 28 pages. http://dx.doi.org/10.1016/0031-9163(65)90885-1 http://dx.doi.org/10.1007/BF02755212 http://dx.doi.org/10.1134/1.1490105 http://dx.doi.org/10.1063/1.1897183 20 S. Post [6] Kalnins E.G., Kress J.M., Miller W. Jr., Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems, J. Math. Phys. 47 (2006), 093501, 25 pages. [7] Kalnins E.G., Kress J.M., Miller W. Jr., Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform, J. Math. Phys. 46 (2005), 053510, 15 pages. [8] Post S., Coupling constant metamorphosis, the Stäckel transform and superintegrability, in Symmetries in Nature: Symposium in Memoriam Marcos Moshinsky (August 9–14, 2010, Cuernavaca, Mexico), Editors L. Benet, P.O. Hess, J.M. Torres and K.B. Wolf, AIP Conf. Proc. 1323 (2011), 265–274. [9] Daskaloyannis C., Ypsilantis K., Unified treatment and classification of superintegrable systems with inte- grals quadratic in momenta on a two dimensional manifold, J. Math. Phys. 47 (2006), 042904, 38 pages, math-ph/0412055. [10] Daskaloyannis C., Ypsilantis K., Quantum superintegrable systems with quadratic integrals on a two di- mensional manifold, J. Math. Phys. 48 (2007), 072108, 22 pages, math-ph/0607058. [11] Kress J.M., Equivalence of superintegrable systems in two dimensions, Phys. Atomic Nuclei 70 (2007), 560–566. [12] Daskaloyannis C., Tanoudis Y., Classification of the quantum two-dimensional superintegrable systems with quadratic integrals and the Stäckel transforms, Phys. Atomic Nuclei 71 (2008), 853–861. [13] Kalnins E.G., Kress J.M., Miller W. Jr., Post S., Structure theory for second order 2D superintegrable systems with 1-parameter potentials, SIGMA 5 (2009), 008, 24 pages, arXiv:0901.3081. [14] Daskaloyannis C., Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems, J. Math. Phys. 42 (2001), 1100–1119, math-ph/0003017. [15] Daskaloyannis C., Tanoudis Y., Ternary Poisson algebra for the nondegenerate three dimensional Kepler– Coulomb potential, in Proceedings of the Fourth International Workshop on Group Analysis of Differential Equations and Integrable Systems (October 26–30, 2008, Protaras, Cyprus), University of Cyprus, Nicosia, 2009, 173–181, arXiv:0902.0259. [16] Kalnins E.G., Miller W. Jr., Post S., Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere, arXiv:1010.3032. [17] Marquette I. Generalized MICZ-Kepler system, duality, polynomial, and deformed oscillator algebras, J. Math. Phys. 51 (2010), 102105, 10 pages, arXiv:1004.4579. [18] Quesne C., Quadratic algebra approach to an exactly solvable position-dependent mass Schrödinger equation in two dimensions, SIGMA 3 (2007), 067, 14 pages, arXiv:0705.2577. [19] Magnus W., Oberhettinger F., Soni R.P., Formula and theorems for the special functions of mathematical physics, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. [20] Kalnins E.G., Miller W. Jr., Post S., Models for quadratic algebras associated with second order superinte- grable systems in 2D, SIGMA 4 (2008), 008, 21 pages, arXiv:0801.2848. [21] Kalnins E.G., Miller W. Jr., Post S., Models of quadratic quantum algebras and their relation to classical superintegrable systems, Phys. Atomic Nuclei 72 (2009), 801–808. [22] Post S., Models of second-order superintegrable systems, PhD Thesis, University of Minnesota, 2009. [23] Kalnins E.G., Miller W. Jr., Post S., Wilson polynomials and the generic superintegrable system on the 2-sphere, J. Phys. A: Math. Theor. 40 (2007), 11525–11538. [24] Kalnins E.G., Miller W. Jr., Reid G.J., Separation of variables for complex Riemannian spaces of constant curvature. I. Orthogonal separable coordinates for SnC and EnC, Proc. Roy. Soc. London Ser. A 39 (1984), 183–206. [25] Kalnins E.G., Kress J.M., Miller W. Jr., Pogosyan G., Nondegenerate superintegrable systems in n- dimensional Euclidean spaces, Phys. Atomic Nuclei 70 (2007), 545–553. [26] Daskaloyannis C., Tanoudis Y., Quadratic algebras for three dimensional non degenerate superintegrable systems with quadratic integrals of motion, arXiv:0902.0130. [27] Kalnins E.G., Miller W. Jr., Post S., Models for the 3D singular isotropic oscillator quadratic algebra, Phys. Atomic Nuclei 73 (2009), 359–366. http://dx.doi.org/10.1063/1.2337849 http://dx.doi.org/10.1063/1.1894985 http://dx.doi.org/10.1063/1.3537855 http://dx.doi.org/10.1063/1.2192967 http://arxiv.org/abs/math-ph/0412055 http://dx.doi.org/10.1063/1.2746132 http://arxiv.org/abs/math-ph/0607058 http://dx.doi.org/10.1134/S1063778807030167 http://dx.doi.org/10.1134/S106377880805013X http://dx.doi.org/10.3842/SIGMA.2009.008 http://arxiv.org/abs/0901.3081 http://dx.doi.org/10.1063/1.1348026 http://arxiv.org/abs/math-ph/0003017 http://arxiv.org/abs/0902.0259 http://arxiv.org/abs/1010.3032 http://dx.doi.org/10.1063/1.3496900 http://arxiv.org/abs/1004.4579 http://dx.doi.org/10.3842/SIGMA.2007.067 http://arxiv.org/abs/0705.2577 http://dx.doi.org/10.3842/SIGMA.2008.008 http://arxiv.org/abs/0801.2848 http://dx.doi.org/10.1134/S1063778809050081 http://dx.doi.org/10.1088/1751-8113/40/38/005 http://dx.doi.org/10.1098/rspa.1984.0075 http://dx.doi.org/10.1134/S1063778807030143 http://arxiv.org/abs/0902.0130 http://dx.doi.org/10.1134/S1063778810020249 http://dx.doi.org/10.1134/S1063778810020249 1 Introduction 2 Exposition of a model: the singular isotropic oscillator 3 Models of quadratic algebras for non-degnerate second-order superintegrable systems 3.1 E2: Smorodinsky-Winternitz ii 3.2 E10 3.3 E15 3.4 E1, Smorodinsky-Winternitz i 3.5 E8 3.6 S9: the generic system in 2D 3.7 E20: Smorodinksy-Winternitz iii 4 2D degenerate systems 4.1 E18: the Coulomb system in 2D 4.2 S3 4.3 E14 4.4 E6 4.5 E5 4.6 E4 5 Conclusion References

Models of Quadratic Algebras Generated by Superintegrable Systems in 2D

Репозитарії

Схожі ресурси