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

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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 Ψₙ and the energies Eₙ of the bound states are explicitly obtained. Finally it is proven that the solutions determine a family of orthogonal polynomials Pₙ(x) related with the Hermite polynomials and such that: (i) Every Pₙ 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.