Shape invariance through Crum transformation

Details Shape invariance through Crum transformation Shape invariance through Crum transformation

2006-03-23

arxiv.org/abs/math-ph/0603056v4

J.Math.Phys.47:122104,2006

Physics

Mathematical Physics

26 pp, one more reference, J.-M. Sparenberg and D. Baye, J. Phys. A 28, 5079 (1995), has been added as Ref. 18 in the publishe

Scientific paper

10.1063/1.2397556

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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