Wilson polynomials and the generic superintegrable system on the 2-sphere

Abstract

For complex two-dimensional Riemannian spaces every classical or quantum second-order superintegrable system can be obtained from a single generic 3-parameter potential on the complex 2-sphere by delicate limit operations and through Stäckel transforms between manifolds. Here we derive families of finite- and infinite-dimensional irreducible representations of the corresponding quadratic quantum algebra for the 2-sphere and point out their role in explaining the degeneracy of the energy eigenspaces corresponding to bound state and continuous spectra of quantum and wave equation analogs of this system. The algebra is exactly the one that describes the Wilson and Racah polynomials in their full generality.