# A quantum quasi-harmonic nonlinear oscillator with an isotonic term

## Abstract

The properties of a nonlinear oscillator with an additional term k{sub g}/x², characterizing the isotonic oscillator, are studied. The nonlinearity affects to both the kinetic term and the potential and combines two nonlinearities associated to two parameters, κ and k{sub g}, in such a way that for κ = 0 all the characteristics of the standard isotonic system are recovered. The first part is devoted to the classical system and the second part to the quantum system. This is a problem of quantization of a system with position-dependent mass of the form m(x) = 1/(1 − κx²), with a κ-dependent non-polynomial rational potential and with an additional isotonic term. The Schrödinger equation is exactly solved and the (κ, k{sub g})-dependent wave functions and bound state energies are explicitly obtained for both κ < 0 and κ > 0.

- Authors:

- Dep. de Física Teórica and IUMA, Universidad de Zaragoza, 50009 Zaragoza (Spain)

- Publication Date:

- OSTI Identifier:
- 22306084

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUND STATE; EXACT SOLUTIONS; NONLINEAR PROBLEMS; OSCILLATORS; POTENTIALS; QUANTIZATION; SCHROEDINGER EQUATION; WAVE FUNCTIONS

### Citation Formats

```
Rañada, Manuel F., E-mail: mfran@unizar.es.
```*A quantum quasi-harmonic nonlinear oscillator with an isotonic term*. United States: N. p., 2014.
Web. doi:10.1063/1.4892084.

```
Rañada, Manuel F., E-mail: mfran@unizar.es.
```*A quantum quasi-harmonic nonlinear oscillator with an isotonic term*. United States. doi:10.1063/1.4892084.

```
Rañada, Manuel F., E-mail: mfran@unizar.es. Fri .
"A quantum quasi-harmonic nonlinear oscillator with an isotonic term". United States.
doi:10.1063/1.4892084.
```

```
@article{osti_22306084,
```

title = {A quantum quasi-harmonic nonlinear oscillator with an isotonic term},

author = {Rañada, Manuel F., E-mail: mfran@unizar.es},

abstractNote = {The properties of a nonlinear oscillator with an additional term k{sub g}/x², characterizing the isotonic oscillator, are studied. The nonlinearity affects to both the kinetic term and the potential and combines two nonlinearities associated to two parameters, κ and k{sub g}, in such a way that for κ = 0 all the characteristics of the standard isotonic system are recovered. The first part is devoted to the classical system and the second part to the quantum system. This is a problem of quantization of a system with position-dependent mass of the form m(x) = 1/(1 − κx²), with a κ-dependent non-polynomial rational potential and with an additional isotonic term. The Schrödinger equation is exactly solved and the (κ, k{sub g})-dependent wave functions and bound state energies are explicitly obtained for both κ < 0 and κ > 0.},

doi = {10.1063/1.4892084},

journal = {Journal of Mathematical Physics},

number = 8,

volume = 55,

place = {United States},

year = {Fri Aug 01 00:00:00 EDT 2014},

month = {Fri Aug 01 00:00:00 EDT 2014}

}