# 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:

- Departamento de Fisica, Universidad de los Andes, Cra.1E No. 18A-10, Santafe de Bogota (Colombia)
- (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}

}

DOI: 10.1063/1.2397556

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