Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials

Abstract

We show explicitly that all 2nd order superintegrable systems in 2 dimensions are limiting cases of a single system: the generic 3-parameter potential on the 2-sphere, S9 in our listing. We extend the Wigner-Inönü method of Lie algebra contractions to contractions of quadratic algebras and show that all of the quadratic symmetry algebras of these systems are contractions of that of S9. Amazingly, all of the relevant contractions of these superintegrable systems on flat space and the sphere are uniquely induced by the well known Lie algebra contractions of e(2) and so(3). By contracting function space realizations of irreducible representations of the S9 algebra (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems, and using Wigner's idea of "saving" a representation, we obtain the full Askey scheme of hypergeometric orthogonal polynomials. This relationship directly ties the polynomials and their structure equations to physical phenomena. It is more general because it applies to all special functions that arise from these systems via separation of variables, not just those of hypergeometric type, and it extends to higher dimensions.

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• Bôcher Contractions of Conformally Superintegrable Laplace Equations
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• Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras
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• Dynamical symmetry algebra of two superintegrable two-dimensional systems
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• Generalized quadratic commutator algebras of PBW-type
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• The universal enveloping algebra of ${\mathfrak{sl}}_2$ and the Racah algebra
• Contractions of Degenerate Quadratic Algebras, Abstract and Geometric
• Lie Algebra Contractions and Separation of Variables on Two-Dimensional Hyperboloids. Coordinate Systems
• Conformal Laplace superintegrable systems in 2D: polynomial invariant subspaces