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A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator. (English) Zbl 1138.81380
Summary: 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 $\Psi n$ 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 ${\cal P}_n(x) $ related to the Hermite polynomials and such that: (i) every ${\cal 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.

81Q05Closed and approximate solutions to quantum-mechanical equations
81U15Exactly and quasi-solvable systems (quantum theory)
33C45Orthogonal polynomials and functions of hypergeometric type
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