On the Eigenfunctions of the Hartley-Bessel Transform: An Approach through Supersymmetric Wigner-Dunkl Quantum Mechanics

Integral transforms, such as the Fourier transform and Hartley transform are indispensable tools in various scientific disciplines, particularly in solving differential and difference-differential equations. In quantum mechanics, these transforms play crucial roles, with the Fourier transform being pivotal in addressing the Schrödinger equation for harmonic oscillators. The Hartley transform, introduced as an alternative to the Fourier transform, shares essential properties and finds applications in supersymmetric quantum mechanics. This paper explores the integration of Wigner-Dunkl Quantum Mechanics with supersymmetric Quantum Mechanics, following early investigations by Post et al. and subsequent studies. We propose a modification to the conventional derivative operator by introducing the Dunkl operator, leading to the derivation of overcomplete bases for Hartley-Bessel transform eigenvectors.