# A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour

## Abstract

The quantum version of a non-linear oscillator, previously analyzed at the classical level, is studied. This is a problem of quantization of a system with position-dependent mass of the form m = (1 + {lambda}x {sup 2}){sup -1} and with a {lambda}-dependent non-polynomial rational potential. This {lambda}-dependent system can be considered as a deformation of the harmonic oscillator in the sense that for {lambda} {sup {yields}} 0 all the characteristics of the linear oscillator are recovered. First, the {lambda}-dependent Schroedinger equation is exactly solved as a Sturm-Liouville problem, and the {lambda}-dependent eigenenergies and eigenfunctions are obtained for both {lambda} > 0 and {lambda} < 0. The {lambda}-dependent wave functions appear as related with a family of orthogonal polynomials that can be considered as {lambda}-deformations of the standard Hermite polynomials. In the second part, the {lambda}-dependent Schroedinger equation is solved by using the Schroedinger factorization method, the theory of intertwined Hamiltonians, and the property of shape invariance as an approach. Finally, the new family of orthogonal polynomials is studied. We prove the existence of a {lambda}-dependent Rodrigues formula, a generating function and {lambda}-dependent recursion relations between polynomials of different orders.

- Authors:

- Departamento de Fisica Teorica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza (Spain). E-mail: jfc@unizar.es
- Departamento de Fisica Teorica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza (Spain). E-mail: mfran@unizar.es
- Departamento de Fisica Teorica, Facultad de Ciencias, Universidad de Valladolid, 47011 Valladolid (Spain). E-mail: msn@fta.uva.es

- Publication Date:

- OSTI Identifier:
- 20976731

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Annals of Physics (New York); Journal Volume: 322; Journal Issue: 2; Other Information: DOI: 10.1016/j.aop.2006.03.005; PII: S0003-4916(06)00072-8; Copyright (c) 2006 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DEFORMATION; EIGENFUNCTIONS; EXACT SOLUTIONS; FACTORIZATION; HAMILTONIANS; HARMONIC OSCILLATORS; HERMITE POLYNOMIALS; MASS; NONLINEAR PROBLEMS; POTENTIALS; QUANTIZATION; RECURSION RELATIONS; SCHROEDINGER EQUATION; WAVE FUNCTIONS

### Citation Formats

```
Carinena, Jose F., Ranada, Manuel F., and Santander, Mariano.
```*A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour*. United States: N. p., 2007.
Web. doi:10.1016/j.aop.2006.03.005.

```
Carinena, Jose F., Ranada, Manuel F., & Santander, Mariano.
```*A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour*. United States. doi:10.1016/j.aop.2006.03.005.

```
Carinena, Jose F., Ranada, Manuel F., and Santander, Mariano. Thu .
"A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour". United States.
doi:10.1016/j.aop.2006.03.005.
```

```
@article{osti_20976731,
```

title = {A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour},

author = {Carinena, Jose F. and Ranada, Manuel F. and Santander, Mariano},

abstractNote = {The quantum version of a non-linear oscillator, previously analyzed at the classical level, is studied. This is a problem of quantization of a system with position-dependent mass of the form m = (1 + {lambda}x {sup 2}){sup -1} and with a {lambda}-dependent non-polynomial rational potential. This {lambda}-dependent system can be considered as a deformation of the harmonic oscillator in the sense that for {lambda} {sup {yields}} 0 all the characteristics of the linear oscillator are recovered. First, the {lambda}-dependent Schroedinger equation is exactly solved as a Sturm-Liouville problem, and the {lambda}-dependent eigenenergies and eigenfunctions are obtained for both {lambda} > 0 and {lambda} < 0. The {lambda}-dependent wave functions appear as related with a family of orthogonal polynomials that can be considered as {lambda}-deformations of the standard Hermite polynomials. In the second part, the {lambda}-dependent Schroedinger equation is solved by using the Schroedinger factorization method, the theory of intertwined Hamiltonians, and the property of shape invariance as an approach. Finally, the new family of orthogonal polynomials is studied. We prove the existence of a {lambda}-dependent Rodrigues formula, a generating function and {lambda}-dependent recursion relations between polynomials of different orders.},

doi = {10.1016/j.aop.2006.03.005},

journal = {Annals of Physics (New York)},

number = 2,

volume = 322,

place = {United States},

year = {Thu Feb 15 00:00:00 EST 2007},

month = {Thu Feb 15 00:00:00 EST 2007}

}