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

TL;DR

The paper solves the radial harmonic oscillator in a position-dependent mass background using a point canonical transformation to map to the PT I potential with constant mass, revealing a deformed shape invariance in a deformed SUSYQM framework. It then constructs rational extensions of the radial oscillator via $X_m$-Jacobi exceptional orthogonal polynomials, obtaining extended potentials whose spectra either remain isospectral (types I/II) or gain an extra bound state (type III). The extended potentials and their wavefunctions are mapped back to the PDM setting, yielding explicit expressions and showcasing deformed SUSY relations that preserve structure for the I and II families; in the α→0 limit, results reduce to known constant-mass Laguerre/EOP cases. This approach provides a unified exactly-solvable framework linking PDM quantum systems, SUSYQM, and exceptional orthogonal polynomials with practical implications for solvable rational extensions. Potential future directions include multi-index extensions and connections to para-Jacobi polynomials.

Abstract

We show that the radial harmonic oscillator problem in the position-dependent mass background of the type $m(α;r) = (1+αr^2)^{-2}$, $α>0$, 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 $α$ goes to zero.

Figures (4)

• Figure 1: Plots of $E^{(\alpha)}_n(L,\omega)$, defined in Eq. (\ref{['eq:E']}), in terms of $\alpha$ for $n=0$, 1, 2, and 3. The parameter values are $L=\omega =1$.
• Figure 2: Plots of the ground-state wavefunction $\psi^{(\alpha)}_0(r;L,\omega)$ in terms of $r$ for $\alpha = 1/\sqrt{3}$ (red line), $\alpha=1/(2\sqrt{2})$ (green line), and $\alpha=0$ (black line). The parameter values are $L=\omega=1$.
• Figure 3: Plots of the extended potential $V^{(1)}_{\rm ext}(r;L,\omega)$ in terms of $r$ for $\alpha=1/\sqrt{3}$ (red line) and $\alpha=0$ (black line). The parameter values are $L=\omega=1$.
• Figure 4: Plots of the wavefunction $\psi^{(\rm ext)}_n(r;L,\omega)$, defined in Eqs. (\ref{['eq:psi-ext-1']})--(\ref{['eq:psi-ext-3']}), in terms of $r$ for $n=0$ (full line), $n=1$ (long-dashed line), and $n=2$ (short-dashed line). The parameter values are $L=\omega=1$ and $\alpha=1/\sqrt{3}$.