Commuting difference operators with polynomial eigenfunctions

TL;DR

This paper constructs explicit commuting difference operators with polynomial eigenfunctions, linking them to q-polynomials and a new integrable system that generalizes Calogero-Moser models.

Contribution

It introduces explicit generators for an algebra of commuting difference operators with polynomial eigenfunctions, connecting to a novel integrable system in mathematical physics.

Findings

Operators are diagonalized by Koornwinder's multivariable q-polynomials

The algebra models quantum integrals of a new difference-type integrable system

System generalizes Calogero-Moser models for non-exceptional root systems

Abstract

We present explicit generators of an algebra of commuting difference operators with trigonometric coefficients. The operators are simultaneously diagonalized by recently discovered q-polynomials (viz. Koornwinder's multivariable generalization of the Askey-Wilson polynomials). From the viewpoint of physics the algebra can be interpreted as consisting of the quantum integrals of a novel difference-type integrable sytem. This system generalizes the Calogero-Moser systems associated with non-exceptional root systems.