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Title: Shape invariance through Crum transformation

Abstract

We show in a rigorous way that Crum's result regarding the equal eigenvalue spectrum of Sturm-Liouville problems can be obtained iteratively by successive Darboux transformations. Furthermore, it can be shown that all neighboring Darboux-transformed potentials of higher order, u{sub k} and u{sub k+1}, satisfy the condition of shape invariance provided the original potential u does so. Based on this result, we prove that under the condition of shape invariance, the nth iteration of the original Sturm-Liouville problem defined solely through the shape invariance is equal to the nth Crum transformation.

Authors:
; ;  [1];  [2]
  1. Departamento de Fisica, Universidad de los Andes, Cra.1E No. 18A-10, Santafe de Bogota (Colombia)
  2. (Mexico)
Publication Date:
OSTI Identifier:
20861550
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 12; Other Information: DOI: 10.1063/1.2397556; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; EIGENVALUES; ITERATIVE METHODS; POTENTIALS; STURM-LIOUVILLE EQUATION; TRANSFORMATIONS

Citation Formats

Organista, Jose Orlando, Nowakowski, Marek, Rosu, H. C., and Potosinian Institute of Science and Technology, Apartado Postal 3-74 Tangamanga, 78231 San Luis Potosi. Shape invariance through Crum transformation. United States: N. p., 2006. Web. doi:10.1063/1.2397556.
Organista, Jose Orlando, Nowakowski, Marek, Rosu, H. C., & Potosinian Institute of Science and Technology, Apartado Postal 3-74 Tangamanga, 78231 San Luis Potosi. Shape invariance through Crum transformation. United States. doi:10.1063/1.2397556.
Organista, Jose Orlando, Nowakowski, Marek, Rosu, H. C., and Potosinian Institute of Science and Technology, Apartado Postal 3-74 Tangamanga, 78231 San Luis Potosi. Fri . "Shape invariance through Crum transformation". United States. doi:10.1063/1.2397556.
@article{osti_20861550,
title = {Shape invariance through Crum transformation},
author = {Organista, Jose Orlando and Nowakowski, Marek and Rosu, H. C. and Potosinian Institute of Science and Technology, Apartado Postal 3-74 Tangamanga, 78231 San Luis Potosi},
abstractNote = {We show in a rigorous way that Crum's result regarding the equal eigenvalue spectrum of Sturm-Liouville problems can be obtained iteratively by successive Darboux transformations. Furthermore, it can be shown that all neighboring Darboux-transformed potentials of higher order, u{sub k} and u{sub k+1}, satisfy the condition of shape invariance provided the original potential u does so. Based on this result, we prove that under the condition of shape invariance, the nth iteration of the original Sturm-Liouville problem defined solely through the shape invariance is equal to the nth Crum transformation.},
doi = {10.1063/1.2397556},
journal = {Journal of Mathematical Physics},
number = 12,
volume = 47,
place = {United States},
year = {Fri Dec 15 00:00:00 EST 2006},
month = {Fri Dec 15 00:00:00 EST 2006}
}