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arXiv:2501.08424 · Journal of Mathematical Physics

Isochronous oscillator with a singular position-dependent mass and its quantization

Aritra Ghosh, Bhabani Prasad Mandal, Bijan Kumar Bagchi

Published Aug 1, 2025• 4 citations
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Abstract

In this paper, we present an analysis of the equation ẍ−(1/2x)ẋ2+2ω2x−1/8x=0, where ω > 0 and x = x(t) is a real-valued variable. We first discuss the appearance of this equation from a position-dependent-mass scenario in which the mass profile goes inversely with x, admitting a singularity at x = 0. The associated potential is also singular at x = 0, splitting the real axis into two halves, i.e., x > 0 and x < 0. The dynamics is exactly solvable for both the branches and so for definiteness, we stick to the x > 0 branch. Performing a canonical quantization in the position representation and upon employing the ordering strategy of the kinetic-energy operator due to von Roos, we show that the problem is isospectral to the isotonic oscillator. Thus, the quantum spectrum consists of an infinite number of equispaced levels. The spacing between the energy levels is found to be insensitive to the specific choices of the ambiguity parameters that are employed for ordering the kinetic-energy operator à la von Roos.

DOI
10.1063/5.0271167

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In this paper, we present an analysis of the equation ẍ−(1/2x)ẋ2+2ω2x−1/8x=0, where ω > 0 and x = x(t) is a real-valued variable. We first discuss the appearance of this equation from a position-dependent-mass scenario in which the mass profile goes inversely with x, admitting a singularity at x = 0. The associated potential is also singular at x = 0, splitting the real axis into two halves, i.e., x > 0 and x < 0. The dynamics is exactly solvable for both the branches and so for definiteness, we stick to the x > 0 branch. Performing a canonical quantization in the position representation and upon employing the ordering strategy of the kinetic-energy operator due to von Roos, we show that the problem is isospectral to the isotonic oscillator. Thus, the quantum spectrum consists of an infinite number of equispaced levels. The spacing between the energy levels is found to be insensitive to the specific choices of the ambiguity parameters that are employed for ordering the kinetic-energy operator à la von Roos.
One of the most important scientific contributions of Professor Marcos Moshinsky has been his study on the harmonic oscillator in quantum theory vis à vis the standard Schrödinger equation with constant mass [1].However, a simple description of the motion of a particle interacting with an external environment such as happen in compositionally graded alloys consist of replacing the mass by the so-called effective mass that is in general variable and dependent on position.Therefore, honoring in memoriam Marcos Moshinsky, in this work we consider the position-dependent mass Schrodinger equations (PDMSE) for the harmonic oscillator potential model as former potential as well as with equi-spaced spectrum solutions, i.e. harmonic oscillator isospectral partners.To that purpose, the point canonical transformation method to convert a general second order differential equation (DE), of Sturm-Liouville type, into a Schrödinger-like standard equation is applied to the PDMSE.In that case, the former potential associated to the PDMSE and the potential involved in the Schrödinger-like standard equation are related through a Riccati-type relationship that includes the equivalent of the Witten superpotential to determine the exactly solvable positions-dependent mass distribution (PDMD) m(x).Even though the proposed approach is exemplified with the harmonic oscillator potential, the procedure is general and can be straightforwardly applied to other DEs.
We revisit the canonical quantization to assess the spectrum of the modified Emden equation x¨+kxx˙+ω2x+k29x3=0\ddot{x} + kx\dot{x} + ω^2 x + \frac{k^2}{9}x^3 = 0x¨+kxx˙+ω2x+9k2​x3=0, which is an isochronous case of the Liénard-Kukles equation. While its classical isochronicity and canonical quantization, leading to polynomial solutions with an exactly-equispaced spectrum have been discussed earlier, including in the recent paper [Int. J. Theor. Phys. 64, 212 (2025)], the present study focuses on the quantization of its branched Hamiltonians. For small kkk, we show numerically that the resulting energy spectrum is no longer perfectly harmonic but only approximately equispaced, exhibiting quasi-harmonic behavior characterized by deviations from uniform spacing. Our numerical results are precisely validated by analytical calculations based on perturbation theory.
We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. 62, 092101 (2021)] in the classical and quantum formalisms by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space (x, p) to a deformed one (xγ, Πγ). Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases.
In this work, the quantum operator approach is applied to both, the position-dependent mass Schrödinger equation (PDMSE) and the Schrodinger equation with constant mass (CMSE). This fact enable us to find the factorization operators that relates both Hamiltonians by means of a kinetic energy operator that comes from the proposal of Morrow and Brownstein. With this approach is possible to find the exactly-solvable PDMSE, for any value of the parameters α and γ in the von Roos's Hamiltonian. For that, our proposal can be considered as a unified treatment of the PDMSE because it contains as particular cases, the kinetic energy operators of various authors such as BenDaniel-Duke, Gora-Williams, Zhu-Kroemer and Li-Kuhn among others. To show the usefulness of our result, we show the solvable PDMSE that comes from the harmonic oscillator potential model for the CMSE. The proposal is general and can easily be extended to other potential models and mass distributions which will be given in the extended paper.
An exactly solvable model of a nonrelativistic quantum singular oscillator with a controlled potential is constructed. A position-dependent mass is introduced instead of a constant mass of the quantum system under consideration in such a way that it is possible to purposefully control the shape of the potential, turning it into a quantum well with different depths. It is shown that analytical expressions of the wave functions of such a quantum system are described through 2F1 type hypergeometric functions and Jacobi polynomials, whereas, the discrete energy spectrum corresponding to these analytic solutions is non-equidistant. In a specific limit, when the above-mentioned control of the potential shape is lost, the wave functions accurately restore the wave functions of an ordinary quantum singular oscillator, which are described through generalized Laguerre polynomials. At this limit, the energy spectrum also restores the usual equidistant energy spectrum of a quantum singular oscillator.