Rational extensions of an oscillator-shaped quantum well potential in a position-dependent mass background

TL;DR

This paper extends an oscillator-shaped quantum well model with position-dependent mass using rational extensions linked to exceptional orthogonal polynomials, resulting in new solvable models with preserved spectra.

Contribution

It introduces rationally-extended position-dependent mass quantum well models based on exceptional orthogonal polynomials, expanding solvable potentials in quantum mechanics.

Findings

Constructed rational extensions with X1-Jacobi polynomials

Developed models with the same spectrum as original potentials

Explored models associated with X2-Jacobi polynomials

Abstract

We show that a recently proposed oscillator-shaped quantum well model associated with a position-dependent mass can be solved by applying a point canonical transformation to the constant-mass Schr\"odinger equation for the Scarf I potential. On using the known rational extension of the latter connected with $X_1$-Jacobi exceptional orthogonal polynomials, we build a rationally-extended position-dependent mass model with the same spectrum as the starting one. Some more involved position-dependent mass models associated with $X_2$-Jacobi exceptional orthogonal polynomials are also considered.