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

TL;DR

This paper demonstrates that all 2D second-order superintegrable systems can be derived as limits of a single 3-parameter system on the 2-sphere, linking their symmetry algebras to the Askey scheme of hypergeometric orthogonal polynomials.

Contribution

It extends Lie algebra contraction methods to quadratic algebras, unifying superintegrable systems and their associated special functions within the Askey scheme.

Findings

All 2D superintegrable systems are contractions of the S9 system.

Contractions of symmetry algebras induce the Askey scheme of polynomials.

The approach applies to all special functions from these systems, beyond hypergeometric types.

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-In\"on\"u 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.