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

2007/11/30 by J. F. Cariñena, A. M. Perelomov, M. F. Rañada +1 · 1 citation
Physics and Astronomy · #quant-ph

paper · pdf · doi:10.1088/1751-8113/41/8/085301

published as J. Phys. A: Math. Theor. 41, 085301 (10 p.) (2008) · 11 pages, 11 figures

arxiv created 2007/11/30 · arxiv updated 2010/11/16

Abstract

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 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 \cal P_n(x) related with 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.

Cited by

• Relativistic and nonrelativistic bound states of the isotonic oscillator by Nikiforov-Uvarov method

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