Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials

TL;DR

This paper demonstrates that equilibrium positions in certain integrable systems are described by Askey-Wilson polynomials, and their quantum versions are shape invariant with eigenfunctions also given by these polynomials.

Contribution

It extends previous work by linking the equilibrium positions of Ruijsenaars-Schneider-van Diejen systems to Askey-Wilson polynomials and shows their quantum versions are shape invariant.

Findings

Equilibrium positions are zeros of Askey-Wilson polynomials.

Quantum systems are shape invariant with eigenfunctions as Askey-Wilson polynomials.

Generalizes known results from Hermite and Laguerre polynomials to Askey-Wilson polynomials.

Abstract

We show that the equilibrium positions of the Ruijsenaars-Schneider-van Diejen systems with the trigonometric potential are given by the zeros of the Askey-Wilson polynomials with five parameters. The corresponding single particle quantum version, which is a typical example of "discrete" quantum mechanical systems with a q-shift type kinetic term, is shape invariant and the eigenfunctions are the Askey-Wilson polynomials. This is an extension of our previous study [1,2], which established the "discrete analogue" of the well-known fact; The equilibrium positions of the Calogero systems are described by the Hermite and Laguerre polynomials, whereas the corresponding single particle quantum versions are shape invariant and the eigenfunctions are the Hermite and Laguerre polynomials.