Shape Invariant Potentials in "Discrete Quantum Mechanics"

TL;DR

This paper explores shape invariant potentials within discrete quantum mechanics, focusing on their role in exactly solvable models related to Ruijsenaars-Schneider systems and associated special functions.

Contribution

It introduces and discusses new examples of shape invariant discrete quantum systems linked to integrable multi-particle models and special polynomials.

Findings

Identification of shape invariant systems related to Ruijsenaars-Schneider models

Connection between these systems and deformed Hermite and Laguerre polynomials

Enhanced understanding of exactly solvable discrete quantum models

Abstract

Shape invariance is an important ingredient of many exactly solvable quantum mechanics. Several examples of shape invariant ``discrete quantum mechanical systems" are introduced and discussed in some detail. They arise in the problem of describing the equilibrium positions of Ruijsenaars-Schneider type systems, which are "discrete" counterparts of Calogero and Sutherland systems, the celebrated exactly solvable multi-particle dynamics. Deformed Hermite and Laguerre polynomials are the typical examples of the eigenfunctions of the above shape invariant discrete quantum mechanical systems.