Exact solution of the position-dependent mass Schr\"odinger equation with the completely positive oscillator-shaped quantum well potential

TL;DR

This paper presents two exactly solvable models of the position-dependent mass Schrödinger equation with oscillator-shaped quantum well potentials, analyzing their spectra and wavefunctions expressed through special polynomials.

Contribution

It introduces new exactly solvable quantum well models with position-dependent mass, providing explicit solutions and spectral properties.

Findings

Discrete energy spectra depend on confinement parameters.

Spectrum is equidistant for one-wall confinement, non-equidistant for two-wall confinement.

Wavefunctions involve Laguerre and Jacobi polynomials.

Abstract

Two exactly-solvable confined models of the completely positive oscillator-shaped quantum well are proposed. Exact solutions of the position-dependent mass Schr\"odinger equation corresponding to the proposed quantum well potentials are presented. It is shown that the discrete energy spectrum expressions of both models depend on certain positive confinement parameters. The spectrum exhibits positive equidistant behavior for the model confined only with one infinitely high wall and non-equidistant behavior for the model confined with the infinitely high wall from both sides. Wavefunctions of the stationary states of the models under construction are expressed through the Laguerre and Jacobi polynomials. In general, the Jacobi polynomials appearing in wavefunctions depend on parameters $a$ and $b$, but the Laguerre polynomials depend only on the parameter $a$. Some limits and special cases of the constructed models are discussed.