A Quantum Exactly Solvable Nonlinear Oscillator with quasi-Harmonic Behaviour

Details A Quantum Exactly Solvable Nonlinear Oscillator with quasi-Harmonic Behaviour A Quantum Exactly Solvable Nonlinear Oscillator with quasi-Harmonic Behaviour

2006-04-05

arxiv.org/abs/math-ph/0604008v1

AnnalsPhys.322:434-459,2007

Physics

Mathematical Physics

29 pages, 4 figures

Scientific paper

10.1016/j.aop.2006.03.005

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.

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