Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems

Abstract

This paper is the conclusion of a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in conformally flat spaces. For two-dimensional and for conformally flat three-dimensional spaces with nondegenerate potentials we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension. We also correct an error in an earlier paper in the series ͑that does not alter the structure results͒ and we elucidate the distinction between superintegrable systems with bases of functionally linearly independent and functionally linearly dependent symmetries.

Quadratic Algebra Approach to an Exactly Solvable Position-Dependent Mass Schrödinger Equation in Two Dimensions

“…Such a quadratic algebra closes at level 6 [47] or, in other words, it has a Casimir operator which is a sixth-order differential operator [36],…”

“…The study and classification of such systems, dating back to the 19th century and revived in the 1960ties [27,28,29], have recently been the subject of intense research activities and substantial progress has been made in this area (see [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48] and references quoted therein). In particular, it has been shown that their integrals of motion generate a quadratic Poisson algebra (in the classical case) or a quadratic associative algebra (in the quantum one) with a Casimir of sixth degree in the momenta and the general form of these algebras has been uncovered [36,45,46,47,48]. Algebras of this kind have many similarities to the quadratic Racah algebra QR(3) (a special case of the quadratic Askey-Wilson algebra QAW(3)) [31,32].…”

Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D

“…We first describe how these classical models arise out of standard Hamilton-Jacobi theory. In [44,27] we have shown that for second order superintegrable systems in two dimensions there is a 1-1 relationship between classical quadratic algebras and quantum quadratic algebras, even though these algebras are not isomorphic. In this sense the quantum quadratic algebra, the spectral theory for its irreducible representations and its possible one variable models are already uniquely determined by the classical system.…”

“…For these examples m = 2 and the most complete classification and structure results are known for the second order case. There is an extensive literature on the subject [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22], with a recent new burst of activity [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]. All such systems have been classified for real and complex Riemannian spaces with n = 2 and their associated quadratic algebras of symmetries computed [15,29,30,24,31].…”

Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature

“…More recently, the study of 2D and 3D superintegrable systems on spaces with variable curvature has been addressed [23,24,25,26,27,28,29]. The aim of this paper is to give a general setting, based on quantum deformations, for the explicit construction of certain classes of superintegrable systems on N D spaces with variable curvature.…”