A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator

A nonpolynomial one-dimensional quantum potential representing an oscillator, which 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 on 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 wavefunctions Ψ 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 related to the Hermite polynomials and such that: (i) every 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.