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Physica Scripta

Comment on: ‘PT-/non-PT-symmetric and non-Hermitian Hellmann potential: approximate bound and scattering states with anyℓ-values’

Paolo Amore, Francisco Marcelo Fernández

Published Jun 16, 2015• 0 citations

Abstract

We carry out an analysis of the paper published in Arda and Sever (2014 Phys. Scr. 89 105204 ) and reveal several unclear points. In the first place, the derivation of the main equations does not appear to be correct. In the second place, the expression for the energy does not appear to yield the results reported by the authors. In the third place, they failed to indicate the references reporting the eigenvalues chosen for comparison. In the fourth place the eigenvalues reported by the authors do not show the same order as those of the Hellmann potential thus leading to a different underlying physics. What is more: the spectrum of the modified model is qualitatively different from the one supported by the Hellmann potential.

DOI: 10.1088/0031-8949/90/8/087001

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We carry out an analysis of the paper published in Arda and Sever (2014 Phys. Scr. 89 105204 ) and reveal several unclear points. In the first place, the derivation of the main equations does not appear to be correct. In the second place, the expression for the energy does not appear to yield the results reported by the authors. In the third place, they failed to indicate the references reporting the eigenvalues chosen for comparison. In the fourth place the eigenvalues reported by the authors do not show the same order as those of the Hellmann potential thus leading to a different underlying physics. What is more: the spectrum of the modified model is qualitatively different from the one supported by the Hellmann potential.

We investigate the approximate bound state solutions of the Schrödinger equation for the PT-/non-PT-symmetric and non Hermitian Hellmann potential. Exact energy eigenvalues and corresponding normalized wave functions are obtained. Numerical values of energy eigenvalues for the bound states are compared with the ones obtained before. Scattering state solutions are also studied. Phase shifts of the potential are written in terms of the angular momentum quantum number

\ell

.

We investigate the approximate bound state solutions of the Schr\"odinger equation for the PT-/non-PT-symmetric and non Hermitian Hellmann potential. Exact energy eigenvalues and corresponding normalized wave functions are obtained. Numerical values of energy eigenvalues for the bound states are compared with the ones obtained before. Scattering state solutions are also studied. Phase shifts of the potential are written in terms of the angular momentum quantum number

\ell

.

We show that the authors of the commented paper (Bowen et al 2012 Phys. Scr. 85 065005) draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In some of the studied examples, the author missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operators 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-x plane (Bender 2007 Rep. Prog. Phys. 70 947)

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-

x

plane (Rep. Prog. Phys. {\bf 70} (2007) 947).

Jia and Dutra ( 2006 J. Phys. A: Math. Gen. 39 11877 ) have considered the one- dimensional non-Hermitian complexified potentials with real spectra in the context of position-dependent mass in the Dirac equation. In their second example, a smooth step shape mass distribution is considered and a non-Hermitian non- PT -symmetric Lorentz vector potential is obtained. They have mapped this problem into an exactly solvable Rosen–Morse Schrödinger model and claimed that the energy spectrum is real. The energy spectrum they have reported is pure imaginary or at best forms an empty set. Their claim on the reality of the energy spectrum is fragile, therefore.

Li and Miao [Phys. Rev. A 85, 042110 (2012)] proposed a non-Hermitian Hamiltonian that is neither Hermitian nor

PT

symmetric but exhibits real eigenvalues for some values of the model parameters. In order to explain this fact, they resorted to

PT

-pseudo Hermiticity and to a so-called permutation symmetry. Here we show that the spectrum of this Hamiltonian can be easily analyzed in the usual way in terms of exact or broken antiunitary symmetries that appear to be more relevant than the permutation symmetry. In addition, we show why the authors' Hamiltonian and the well-known Pais-Uhlenbeck oscillator lead to the same fourth-order differential equation for the coordinates.