Orthogonality of Hermite polynomials in superspace and Mehler type formulae

TL;DR

This paper explores Hermite polynomials in superspace with various symmetries, establishing their orthogonality properties, extending classical formulas, and deriving a super Fourier transform eigenfunction decomposition.

Contribution

It introduces a new inner product for superspace Hermite polynomials, restoring orthogonality and hermiticity, and extends Mehler formulas to superspace contexts.

Findings

Orthogonality of Hermite polynomials in superspace with Sp(2n) symmetry established.

A new inner product restores orthogonality and hermiticity in superspace.

A Mehler formula for superspace is derived, enabling super Fourier transform analysis.

Abstract

In this paper, Hermite polynomials related to quantum systems with orthogonal O(m)-symmetry, finite reflection group symmetry G < O(m), symplectic symmetry Sp(2n) and superspace symmetry O(m) x Sp(2n) are considered. After an overview of the results for O(m) and G, the orthogonality of the Hermite polynomials related to Sp(2n) is obtained with respect to the Berezin integral. As a consequence, an extension of the Mehler formula for the classical Hermite polynomials to Grassmann algebras is proven. Next, Hermite polynomials in a full superspace with O(m) x Sp(2n) symmetry are considered. It is shown that they are not orthogonal with respect to the canonically defined inner product. However, a new inner product is introduced which behaves correctly with respect to the structure of harmonic polynomials on superspace. This inner product allows to restore the orthogonality of the Hermite polynomials and also restores the hermiticity of a class of Schroedinger operators in superspace. Subsequently, a Mehler formula for the full superspace is obtained, thus yielding an eigenfunction decomposition of the super Fourier transform. Finally, an extensive comparison is made of the results in the different types of symmetry.