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arXiv:2010.07701 · Physica Scripta

Comment on: Harmonic oscillator in an environment with a pointlike defect. (2019 Phys. Scr. 94 125301)

Francisco Marcelo Fernández

Published Dec 16, 2020• 0 citations
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Abstract

Abstract We analyze recent results for a harmonic oscillator in an environment with a pointlike defect. We show that the allowed oscillator frequencies predicted by the authors stem from a misinterpretation of the exact solutions of a conditionally solvable eigenvalue equation. Also the exact eigenvalues derived by those authors are meaningless because they belong to different quantum-mechanical models.

DOI
10.1088/1402-4896/abd1af

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Comment on: Harmonic oscillator in an environment with a pointlike defect. (2019 Phys. Scr. 94 125301)

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We analyze recent results for a harmonic oscillator in an environment with a pointlike defect. We show that the allowed oscillator frequencies predicted by the authors stem from a misinterpretation of the exact solutions of a conditionally solvable eigenvalue equation. Also the exact eigenvalues derived by those authors are meaningless because they belong to different quantum-mechanical models.

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Comment on `Exact solution of the position-dependent effective mass and angular frequency Schrödinger equation: harmonic oscillator model with quantized confinement parameter'

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Results in Physics•Dec 30, 2021•0 citations
Abstract We analyze recent results for a harmonic oscillator in an environment with a pointlike defect. We show that the allowed oscillator frequencies predicted by the authors stem from a misinterpretation of the exact solutions of a conditionally solvable eigenvalue equation. Also the exact eigenvalues derived by those authors are meaningless because they belong to different quantum-mechanical models.
Abstract In a recent paper by Jafarov et al (2020 J. Phys. A 53 485301), a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position, was constructed and it was shown that the confinement parameter gets quantized. By using a point canonical transformation starting from the constant-mass Schrödinger equation for the Rosen–Morse II potential, it is shown here that similar results can be easily obtained without quantizing the confinement parameter. In addition, an extension to a confined shifted harmonic oscillator directly follows from the same point canonical transformation.
In a recent paper by Jafarov, Nagiyev, Oste and Van der Jeugt (2020 {\sl J.\ Phys.\ A} {\bf 53} 485301), a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position, was constructed and it was shown that the confinement parameter gets quantized. By using a point canonical transformation starting from the constant-mass Schrödinger equation for the Rosen-Morse II potential, it is shown here that similar results can be easily obtained without quantizing the confinement parameter. In addition, an extension to a confined shifted harmonic oscillator directly follows from the same point canonical transformation.
We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operator and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-xxx plane (Rep. Prog. Phys. {\bf 70} (2007) 947).