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

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(2))(-1) and with a lambda-dependent non-polynomial rational potential. This -dependent system can be considered as a deformation of the harmonic oscillator in the sense that for lambda -> 0 all the characteristics of the linear oscillator are recovered. First, the lambda-dependent Schrodinger 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 Schrodinger equation is solved by using the Schrodinger 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. (c) 2006 Elsevier Inc. All rights reserved.