Shape invariance through Crum transformation

TL;DR

This paper rigorously demonstrates that Crum transformations can be iteratively derived through Darboux transformations, establishing that shape invariance is preserved across higher-order potentials and linking Crum's method to shape invariance in Sturm-Liouville problems.

Contribution

It provides a rigorous proof connecting Crum transformations with Darboux transformations and shows that shape invariance is maintained through higher-order potentials.

Findings

Crum's eigenvalue spectrum result can be obtained via Darboux transformations.

Higher-order Darboux-transformed potentials satisfy shape invariance.

n-th Crum transformation equals the n-th shape-invariant potential iteration.

Abstract

We show in a rigorous way that Crum's result on equal eigenvalue spectrum of Sturm-Liouville problems can be obtained iteratively by successive Darboux transformations. It can be shown that all neighbouring Darboux-transformed potentials of higher order, u_{k} and u_{k+1}, satisfy the condition of shape invariance provided the original potential u does. We use this result to proof that under the condition of shape invariance the n-th iteration of the original Sturm-Liouville problem defined through shape invariance is equal to the n-th Crum transformation