A quantum exactly solvable non-linear oscillator related with the isotonic oscillator

Abstract

A nonpolynomial one-dimensional quantum potential representing an oscillator, that can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator (harmonic oscillator with a centripetal barrier), is studied. First the general case, that depends of a parameter $a$, is considered and then a particular case is studied with great detail. It is proven that it is Schrödinger solvable and then the wave functions ${\Psi }_n$ and the energies $E_n$ of the bound states are explicitly obtained. Finally it is proven that the solutions determine a family of orthogonal polynomials ${\mathcal{P}}_n(x)$ related with the Hermite polynomials and such that: (i) Every ${\mathcal{P}}_n$ is a linear combination of three Hermite polynomials, and (ii) They are orthogonal with respect to a new measure obtained by modifying the classic Hermite measure.

11 pages, 11 figures

References in corpus (2)

• A Quantum Exactly Solvable Nonlinear Oscillator with quasi-Harmonic Behaviour
• Perturbation method for non-square Hamiltonians and its application to polynomial oscillators

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