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Zbl 1138.81380
Cariñena, J.F.; Perelomov, A.M.; Rañada, M.F.; Santander, M.
A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator.
(English)
[J] J. Phys. A, Math. Theor. 41, No. 8, Article ID 085301, 10 p. (2008). ISSN 1751-8113; ISSN 1751-8121/e

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.
MSC 2000:
*81Q05 Closed and approximate solutions to quantum-mechanical equations
81U15 Exactly and quasi-solvable systems
33C45 Orthogonal polynomials and functions of hypergeometric type

Cited in: Zbl 1200.81084 Zbl 1177.81034

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