Calogero-Sutherland-Moser Systems, Ruijsenaars-Schneider-van Diejen Systems and Orthogonal Polynomials

TL;DR

This paper explores the connection between integrable multi-particle systems and orthogonal polynomials, showing that the equilibrium positions and eigenfunctions are described by classical and basic hypergeometric orthogonal polynomials, including Hermite, Laguerre, Jacobi, Wilson, and Askey-Wilson.

Contribution

It demonstrates that the orthogonal polynomial structure extends from Calogero-Sutherland-Moser systems to Ruijsenaars-Schneider-van Diejen systems, revealing new links in integrable systems and special functions.

Findings

Equilibrium positions are described by classical orthogonal polynomials.

Eigenfunctions of quantum systems are expressed in terms of hypergeometric polynomials.

RSvD systems inherit polynomial structures, linking to the Askey-scheme.

Abstract

The equilibrium positions of the multi-particle classical Calogero-Sutherland-Moser (CSM) systems with rational/trigonometric potentials associated with the classical root systems are described by the classical orthogonal polynomials; the Hermite, Laguerre and Jacobi polynomials. The eigenfunctions of the corresponding single-particle quantum CSM systems are also expressed in terms of the same orthogonal polynomials. We show that this interesting property is inherited by the Ruijsenaars-Schneider-van Diejen (RSvD) systems, which are integrable deformation of the CSM systems; the equilibrium positions of the multi-particle classical RSvD systems and the eigenfunctions of the corresponding single-particle quantum RSvD systems are described by the same orthogonal polynomials, the continuous Hahn (special case), Wilson and Askey-Wilson polynomials. They belong to the Askey-scheme of the basic hypergeometric orthogonal polynomials and are deformation of the Hermite, Laguerre and Jacobi polynomials, respectively. The Hamiltonians of these single-particle quantum mechanical systems have two remarkable properties, factorization and shape invariance.