Exact solutions of a nonpolynomial oscillator related to isotonic oscillator

Qian Dong; Guo-Hua Sun; N. Saad; Shi-Hai Dong

doi:10.1140/epjp/i2019-12980-7

EPJ

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2024 Impact factor 2.9

EPJ PLUS - Condensed Matter and Complex Systems

Eur. Phys. J. Plus (2019) 134: 562
https://doi.org/10.1140/epjp/i2019-12980-7

Regular Article

Exact solutions of a nonpolynomial oscillator related to isotonic oscillator

Qian Dong¹, Guo-Hua Sun², N. Saad³ and Shi-Hai Dong¹^*

¹ Laboratorio de Información Cuántica, CIDETEC, Instituto Politécnico Nacional, UPALM, CDMX, 07700, Mexico City, Mexico
² Catedrática CONACYT, CIC, Instituto Politécnico Nacional, CDMX 07700, Mexico City, Mexico
³ Department of Mathematics and Statistics, University of Prince Edward Island, 550 University Avenue, C1A 4P3, Charlottetown, PEI, Canada

^* e-mail: dongsh2@yahoo.com

Received: 21 June 2019
Accepted: 1 September 2019
Published online: 12 November 2019

Abstract

We find that the analytical solutions to a quantum system with a nonpolynomial oscillator potential related to isotonic oscillator are given by the confluent Heun functions $H_{c}(\alpha, \beta, \gamma, \delta, \eta; z)$ . The properties of the wave functions, which are strongly relevant for the potential parameters a and g, are illustrated. It is shown that the wave functions are shrunk to the origin for a given a when the potential parameter g increases, while the wave peak of wave functions is concaved to the origin when the negative potential parameter $\vert g\vert$ increases. Moreover, the wave peaks of the even wave functions become sharper when the potential parameter $ a < 1$ decreases, but they become flat when the potential parameter $ a > 1$ increases. When the minimum value $V_{\min}=-g/a^2$ tends to zero, this nonpolynomial oscillator reduces to a harmonic oscillator.