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arXiv:2512.16510

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

Christiane Quesne

math-phnlin.SIquant-phSubmitted Dec 18, 2025• Announced Dec 19, 2025
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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.

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25 pages, 4 figures

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We show that the Scarf I potential problem in a position-dependent mass background of the type m(α;x)=(1+αsin⁡x)−2m(α;x) = (1 + α\sin x)^{-2}m(α;x)=(1+αsinx)−2, 0<α<10<α<10<α<1, can be solved by using a point canonical transformation mapping the corresponding Schr\" odinger equation onto that of the Scarf I potential with constant mass. The inverse point canonical transformation then provides some exactly-solvable rational extensions of the Scarf I potential with positive-dependent mass associated with XmX_mXm​-Jacobi exceptional orthogonal polynomials of type I, II, or III. The Scarf I potential problem with position-dependent mass is shown to exhibit a deformed shape invariance property in a deformed supersymmetric framework. Such a property is also valid for extended potentials of type I and II. The results are illustrated with a simple example.
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