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

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

2007-11-30

arxiv.org/abs/0711.4899v1

J. Phys. A: Math. Theor. 41, 085301 (10 p.) (2008)

Physics

Quantum Physics

11 pages, 11 figures

Scientific paper

10.1088/1751-8113/41/8/085301

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 Schr\"odinger solvable and then the wave functions $\Psi_n$ 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 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.

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