Factorization solution of Cari\~nena's quantum nonlinear oscillator

TL;DR

This paper simplifies the solution of Cariñena's quantum nonlinear oscillator by revealing its supersymmetric relationship to the harmonic oscillator, enabling easier derivation of wavefunctions and energies, and extends this approach to generate new solvable potentials.

Contribution

It demonstrates that Cariñena's potential is a supersymmetric partner of the harmonic oscillator, providing a simpler solution method and generating a family of exactly solvable potentials.

Findings

The potential is a supersymmetric partner of the harmonic oscillator.

Wavefunctions and energies can be derived more simply using supersymmetry.

An infinite set of exactly solvable potentials is generated.

Abstract

In a recent paper Cari\~nena et al analyzed a non-polynomial one-dimensional quantum potential representing an oscillator which they argued was intermediate between the harmonic and isotonic oscillators. In particular they proved that it is Schr\"odinger soluble, and explicitly obtained the wavefunctions and energies of the bound states. In this paper we show that these results can be obtained much more simply by noting that this potential is a supersymmetric partner potential of the harmonic oscillator. We then use this observation to generate an infinite set of potentials which can be exactly solved in a similar manner.