Equilibrium Positions and Eigenfunctions of Shape Invariant (`Discrete') Quantum Mechanics

TL;DR

This paper reviews how certain integrable quantum systems' equilibrium positions and eigenfunctions are described by orthogonal polynomials, highlighting their factorization and shape invariance properties.

Contribution

It elucidates the connection between classical equilibrium positions and quantum eigenfunctions via orthogonal polynomials, emphasizing shape invariance in these systems.

Findings

Equilibrium positions are described by classical orthogonal polynomials.

Eigenfunctions of quantum systems correspond to the same orthogonal polynomials.

Hamiltonians exhibit factorization and shape invariance properties.

Abstract

Certain aspects of the integrability/solvability of the Calogero-Sutherland-Moser systems and the Ruijsenaars-Schneider-van Diejen systems with rational and trigonometric potentials are reviewed. The equilibrium positions of classical multi-particle systems and the eigenfunctions of single-particle quantum mechanics are described by the same orthogonal polynomials: the Hermite, Laguerre, Jacobi, continuous Hahn, Wilson and Askey-Wilson polynomials. The Hamiltonians of these single-particle quantum mechanical systems have two remarkable properties, factorization and shape invariance.