Shape Invariant Potentials in Higher Dimensions

TL;DR

This paper explores the extension of shape invariance properties of quantum potentials from one dimension to higher dimensions, proposing a new ansatz and reformulation that could broaden applications in quantum mechanics.

Contribution

It introduces a simple ansatz for constructing shape invariant potentials in arbitrary dimensions and suggests generalizations for Hamiltonians related through space-time transformations.

Findings

Reconstructed all known 1D shape invariant potentials

Extended the ansatz to higher dimensions

Proposed reformulation and potential generalizations

Abstract

In this paper we investigate the shape invariance property of a potential in one dimension. We show that a simple ansatz allows us to reconstruct all the known shape invariant potentials in one dimension. This ansatz can be easily extended to arrive at a large class of new shape invariant potentials in arbitrary dimensions. A reformulation of the shape invariance property and possible generalizations are proposed. These may lead to an important extension of the shape invariance property to Hamiltonians that are related to standard potential problems via space time transformations, which are found useful in path integral formulation of quantum mechanics.