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

Abstract

We show that the radial harmonic oscillator problem in the position-dependent mass background of the type $m(\hat{I}\pm ;r)=(1+\hat{I}\pm r^2{)}^{-2}$, $\hat{I}\pm >0$, can be solved by using a point canonical transformation mapping the corresponding Schrödinger equation onto that of the Pöschl-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 $\hat{I}\pm$ goes to zero.

25 pages, 4 figures