A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator

Abstract

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