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arXiv · 2512.16510

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

Christiane Quesne
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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.

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Christiane Quesne. 2025-12-18. Rationally-extended radial harmonic oscillator in a position-dependent mass background. https://arxiv.org/abs/2512.16510

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