Abstract
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 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 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.
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