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Shape invariance through Crum transformation

Jose Orlando Organista·M. Nowakowski·H. C. Rosu
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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

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Jose Orlando Organista, M. Nowakowski, H. C. Rosu. 2006-12-11. Shape invariance through Crum transformation. https://doi.org/10.1063/1.2397556

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