Willard Miller (University of Minnesota, USA)

Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere

Abstract:
   We give a brief review of the definition and structure theory for 2nd order quantum superintegrable systems in 2D and 3D conformally flat spaces and describe the associated quadratic symmetry algebras. The representation theory of these symmetry algebras gives important information about the the energy eigenvalues and the spectra of the symmetries. We study realizations of the possible irreducible representations of the quadratic algebras by differential or difference operators in 1 or 2 complex variable acting on Hilbert spaces of analytic functions. These models greatly simplify the study of the representations and are also of considerable interest in their own right. In particular the Wilson and Racah polynomials in 1 and 2 variables emerge naturally in their full generality. This indicates a general relationship between 2nd order superintegrable systems in n dimensions and multivariable orthogonal polynomials.
   (Joint work with E.G. Kalnins and S. Post.)