Potentials with Two Shifted Sets of Equally Spaced Eigenvalues and Their Calogero Spectrum

TL;DR

This paper introduces new quantum potentials inspired by shape invariance, featuring two shifted sets of equally spaced energy levels with unique singularities, expanding the understanding of spectral properties in supersymmetric quantum mechanics.

Contribution

It presents a novel class of potentials with two shifted eigenvalue sets, incorporating singularities and delta-function features, inspired by the Calogero-Sutherland model and shape invariance.

Findings

Potentials with two shifted equally spaced eigenvalue sets identified.

Spectral properties influenced by singularity parameter range.

Potential connections to supersymmetric quantum mechanics explored.

Abstract

Motivated by the concept of shape invariance in supersymmetric quantum mechanics, we obtain potentials whose spectrum consists of two shifted sets of equally spaced energy levels. These potentials are similar to the Calogero-Sutherland model except the singular term $\alpha x^{-2}$ always falls in the transition region $-1/4 < \alpha < 3/4$ and there is a delta-function singularity at x=0.