Rationally-extended radial harmonic oscillator in a position-dependent mass background

TL;DR

This paper solves a position-dependent mass radial harmonic oscillator problem using a point canonical transformation, revealing deformed shape invariance and providing exactly-solvable rational extensions linked to exceptional orthogonal polynomials.

Contribution

It introduces a novel method to solve the radial harmonic oscillator with position-dependent mass and constructs new exactly-solvable extensions with deformed shape invariance.

Findings

Exact solutions for the P"oschl-Teller I potential with position-dependent mass.

Construction of rational extensions using exceptional orthogonal polynomials.

Demonstration of deformed shape invariance in extended potentials.

Abstract

We show that the radial harmonic oscillator problem in the position-dependent mass background of the type $m(\alpha;r) = (1+\alpha r^2)^{-2}$, $\alpha>0$, can be solved by using a point canonical transformation mapping the corresponding Schr\"odinger equation onto that of the P\"oschl-Teller I potential with constant mass. The radial harmonic oscillator problem with position-dependent mass is shown to exhibit a deformed shape invariance property in a deformed supersymmetric framework. The inverse point canonical transformation then provides some exactly-solvable rational extensions of the radial harmonic oscillator with position-dependent mass associated with $X_m$-Jacobi exceptional orthogonal polynomials of type I, II, or III. The extended potentials of type I and II are proved to display deformed shape invariance. The spectrum and wavefunctions of the radial harmonic oscillator potential and its extensions are shown to go over to well-known results when the deforming parameter $\alpha$ goes to zero.