Supersymmetric Quantum Mechanics with Reflections

TL;DR

This paper introduces a supersymmetric quantum mechanics model using differential-difference operators with reflections, featuring an extended Scarf I potential and eigenfunctions related to little -1 Jacobi polynomials, connecting to Dunkl operators.

Contribution

It presents a novel realization of supersymmetric quantum mechanics with reflections, including explicit eigenfunctions and intertwining operators for the extended Scarf I potential.

Findings

Eigenfunctions are expressed in terms of little -1 Jacobi polynomials.

Eigenvalue equations involve Dunkl-type operators.

Intertwining operators relate wave functions with different parameters.

Abstract

We consider a realization of supersymmetric quantum mechanics where supercharges are differential-difference operators with reflections. A supersymmetric system with an extended Scarf I potential is presented and analyzed. Its eigenfunctions are given in terms of little -1 Jacobi polynomials which obey an eigenvalue equation of Dunkl type and arise as a q-> -1 limit of the little q-Jacobi polynomials. Intertwining operators connecting the wave functions of extended Scarf I potentials with different parameters are presented.