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arXiv:2606.23264

Jacobi exceptional orthogonal polynomials for extended Scarf I potentials with position-dependent mass

Christiane Quesne

math-phquant-phSubmitted Jun 22, 2026• Announced Jun 23, 2026
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Abstract

We show that the Scarf I potential problem in a position-dependent mass background of the type $m(α;x) = (1 + α\sin x)^{-2}$, $0<α<1$, 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 $X_m$-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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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.
We show that the radial harmonic oscillator problem in the position-dependent mass background of the type m(α;r)=(1+αr2)−2m(α;r) = (1+αr^2)^{-2}m(α;r)=(1+αr2)−2, α>0α>0α>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 XmX_mXm​-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 ααα goes to zero.
Using shape invariance property we construct coherent state for a class of potentials containing Scarf I and its extensions, the solutions of the latter being given in terms of the recently discovered Jacobi-Xl type exceptional polynomials. It is shown that the coherent state possesses the property of resolution of unity and exhibits sub-Poissonian behavior. We then investigate the coherent state of Scarf I potential and its l = 1 extension in some detail to understand the similarities (and differences) between the exceptional orthogonal polynomials and their classical counterparts.
Abstract The exactly solvable Position Dependent Mass Schrödinger Equation (PDMSE) for Mie-type potentials is presented. To that, by means of a point canonical transformation the exactly solvable constant mass Schrödinger equation is transformed into a PDMSE. The mapping between both Schrödinger equations lets obtain the energy spectra and wave functions for the potential under study. This happens for any selection of the O von Roos ambiguity parameters involved in the kinetic energy operator. The exactly solvable multiparameter exponential-type potential for the constant mass Schrödinger equation constitutes the reference problem allowing to solve the PDMSE for Mie potentials and mass functions of the form given by m(x) = skx s-1/(xs + 1))2. Thereby, as a useful application of our proposal, the particular Lennard-Jones potential is presented as an example of Mie potential by considering the mass distribution m(x) = 6kx 5/(x 6 + 1))2. The proposed method is general and can be straightforwardly applied to the solution of the PDMSE for other potential models and/or with different position-dependent mass distributions.
The revival structures for the X_m exceptional orthogonal polynomials of the Scarf I potential endowed with position-dependent effective mass is studied in the context of the generalized Gazeau-Klauder coherent states. It is shown that in the case of the constant mass, the deduced coherent states mimic full and fractional revivals phenomena. However in the case of position-dependent effective mass, although full revivals take place during their time evolution, there is no fractional revivals as defined in the common sense. These properties are illustrated numerically by means of some specific profile mass functions, with and without singularities. We have also observed a close connection between the coherence time τ_coh^m? and the mass parameter ?.
On using the known equivalence between the presence of a position-dependent mass (PDM) in the Schrödinger equation and a deformation of the canonical commutation relations, a method based on deformed shape invariance has recently been devised for generating pairs of potential and PDM for which the Schrödinger equation is exactly solvable.This approach has provided the bound-state energy spectrum, as well as the ground-state and the first few excited-state wavefunctions.The general wavefunctions have however remained unknown in explicit form because for their determination one would need the solutions of a rather tricky differential-difference equation.Here we show that solving this equation may be avoided by combining the deformed shape invariance technique with the point canonical transformation method in a novel way.It consists in employing our previous knowledge of the PDM problem energy spectrum to construct a constant-mass Schrödinger equation with similar characteristics and in deducing the PDM wavefunctions from the known constantmass ones.Finally, the equivalence of the wavefunctions coming from both approaches is checked.