Mehta's eigenvectors for the finite Hartely transform

TL;DR

This paper introduces a novel method using supersymmetric quantum mechanics to analytically evaluate eigenfunctions of the finite Hartley transform, revealing new eigenvector bases expressed through supersymmetric Hermite polynomials.

Contribution

It presents a new approach leveraging supersymmetric quantum mechanics to find eigenfunctions of the finite Hartley transform, which was not previously achieved.

Findings

Eigenfunctions expressed in terms of supersymmetric Hermite polynomials

New overcomplete basis of eigenvectors for the finite Hartley transform

Demonstrates commutation of the Hartley transform with a supercharge operator

Abstract

This paper presents a novel approach for evaluating analytical eigenfunctions of the finite Hartley transform. The approach is based on the use of $N=1/2$-supersymmetric quantum mechanics as a fundamental tool, which builds on the key observation that the Hartley transform commutes with the supercharge operator. Using the intertwining operator between the Hartley transform and the finite Hartley transform, our approach provides an overcomplete basis of eigenvectors expressed in terms of supersymmetric Hermite polynomials.