Generalized Hermite polynomials in superspace as eigenfunctions of the supersymmetric rational CMS model

TL;DR

This paper constructs orthogonal eigenfunctions for a supersymmetric quantum model, extending Hermite polynomials into superspace, and reveals their relation to Jack superpolynomials, demonstrating the model's maximal superintegrability.

Contribution

It introduces an algebraic method to derive superspace Hermite polynomials as eigenfunctions of the supersymmetric rational CMS model, linking them to Jack superpolynomials.

Findings

Eigenfunctions expressed as differential operators on Jack superpolynomials

Supersymmetric model shown to be maximally superintegrable

Simple relation between Hermite and Jack superpolynomials

Abstract

We present an algebraic construction of the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement. These eigenfunctions are the superspace extension of the generalized Hermite (or Hi-Jack) polynomials. The conserved quantities of the rational supersymmetric model are related to their trigonometric relatives through a similarity transformation. This leads to a simple expression between the corresponding eigenfunctions: the generalized Hermite superpolynomials are written as a differential operator acting on the corresponding Jack superpolynomials. As an aside, the maximal superintegrability of the supersymmetric rational Calogero-Moser-Sutherland model is demonstrated.