A Quantum Exactly Solvable Nonlinear Oscillator with quasi-Harmonic Behaviour

TL;DR

This paper presents an exact quantum solution for a nonlinear oscillator with position-dependent mass, revealing a family of deformed Hermite polynomials and demonstrating the system's quasi-harmonic behavior through analytical methods.

Contribution

It introduces a solvable quantum nonlinear oscillator with a position-dependent mass and develops a new family of orthogonal polynomials as its eigenfunctions.

Findings

Exact eigenenergies and eigenfunctions obtained for all >0 and <0

Eigenfunctions related to -deformations of Hermite polynomials

Established Rodrigues formula, generating function, and recursion relations for the new polynomials

Abstract

The quantum version of a non-linear oscillator, previouly analyzed at the classical level, is studied. This is a problem of quantization of a system with position-dependent mass of the form $m={(1+\lambda x^2)}^{-1}$ and with a $\la$-dependent nonpolynomial rational potential. This $\la$-dependent system can be considered as a deformation of the harmonic oscillator in the sense that for $\la\to 0$ all the characteristics of the linear oscillator are recovered. Firstly, the $\la$-dependent Schr\"odinger equation is exactly solved as a Sturm-Liouville problem and the $\la$-dependent eigenenergies and eigenfunctions are obtained for both $\la>0$ and $\la<0$. The $\la$-dependent wave functions appear as related with a family of orthogonal polynomials that can be considered as $\la$-deformations of the standard Hermite polynomials. In the second part, the $\la$-dependent Schr\"odinger equation is solved by using the Schr\"odinger 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 $\la$-dependent Rodrigues formula, a generating function and $\la$-dependent recursion relations between polynomials of different orders.