Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Comment on “On the consistency of the solutions of the space fractional Schrödinger equation” [J. Math. Phys. 53, 042105 (2012)]

Selçuk Ş. Bayın

Published Jul 1, 2013• 11 citations
Publisher

Abstract

Hawkins and Schwarz [J. Math. Phys. 54, 014101 (2013)]10.1063/1.4772533 claim that there is a flaw in the way certain integrals are evaluated to show the consistency of the infinite square well solutions of the space fractional Schrödinger equation [S. S. Bayin, J. Math. Phys. 53, 042105 (2012)10.1063/1.4705268; S. S. Bayin, J. Math. Phys. 53, 084101 (2012)]10.1063/1.4739758. We give additional details about how these integrals are evaluated and show that there is no inconsistency for an infinite square well.

DOI
10.1063/1.4816007

Related papers

On the consistency of the solutions of the space fractional Schrödinger equation

Selçuk Ş. Bayın

Recently, it was pointed out that the solutions found in the literature for the space fractional Schrödinger equation in a piecewise manner are wrong, except the case with the delta potential. We re-analyze this problem and show that an exact and a proper treatment of the relevant integral prove otherwise. We also discuss effective potential approach and present a free particle solution for the space and time fractional Schrödinger equation in general coordinates in terms of Fox's H-functions.

arXiv:1203.4556•Journal of Mathematical Physics•Apr 1, 2012•66 citations

Fractional Schrödinger equation for a particle moving in a potential well

Yuri Luchko

In this paper, the fractional Schrödinger equation that contains the quantum Riesz fractional derivative instead of the Laplace operator is revisited for the case of a particle moving in the infinite potential well. In the recent papers [M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Schwarz, “On the nonlocality of the fractional Schrödinger equation,” J. Math. Phys. 51, 062102 (2010)10.1063/1.3430552] and [S. S. Bayin, “On the consistency of the solutions of the space fractional Schrödinger equation,” J. Math. Phys. 53, 042105 (2012)10.1063/1.4705268] published in this journal, controversial opinions regarding solutions to the fractional Schrödinger equation for a particle moving in the infinite potential well that were derived by Laskin [“Fractals and quantum mechanics,” Chaos 10, 780–790 (2000)10.1063/1.1050284] have been given. In this paper, a thorough mathematical treatment of these matters is provided. The problem under consideration is reformulated in terms of three integral equations with the power kernels. Even if the equations look not very complicated, no solution to these equations in explicit form is known. Still, the obtained equations are used to show that the eigenvalues and eigenfunctions of the fractional Schrödinger equation for a particle moving in the infinite potential well given by Laskin [“Fractals and quantum mechanics,” Chaos 10, 780–790 (2000)10.1063/1.1050284] and many other papers by different authors cannot be valid as has been first stated by Jeng et al. [“On the nonlocality of the fractional Schrödinger equation,” J. Math. Phys. 51, 062102 (2010)10.1063/1.3430552].

Journal of Mathematical Physics•Jan 1, 2013•123 citations

Comment on “On the energy levels of a finite square-well potential” [J. Math. Phys. 41, 4551 (2000)]

David L. Aronstein, Carlos R. Stroud

Journal of Mathematical Physics•Dec 1, 2000•8 citations

Reply to “Comment on ‘Solution of the Schrödinger equation for the time-dependent linear potential’ ”

I. Guedes

In this reply I discuss further the misconception in my paper [Phys. Rev. A 63, 034102 (2001)] related to the solution of the Schr\"odinger equation for a particle in a time-dependent linear potential.

Physical Review A•Jul 8, 2003•18 citations

Further comment on ‘‘Optimization of approximate solutions to the time-dependent Schrödinger equation’’

José M. Maidagan, Rubén Darío Piacentini

We show that the Reply of Weglein [Phys. Rev. A 31, 4025 (1985)] to our previous Comment [Phys. Rev. A 31, 4023 (1985)] is not valid.Received 18 September 1985DOI:https://doi.org/10.1103/PhysRevA.35.933©1987 American Physical Society

Physical Review A•Jan 1, 1987•0 citations

One-dimensional hydrogen atom in an infinite square well

I. Richard Lapidus

The effect of boundary conditions on solutions to the Schrödinger Equation is demonstrated.

American Journal of Physics•Jun 1, 1982•10 citations
Hawkins and Schwarz [J. Math. Phys. 54, 014101 (2013)]10.1063/1.4772533 claim that there is a flaw in the way certain integrals are evaluated to show the consistency of the infinite square well solutions of the space fractional Schrödinger equation [S. S. Bayin, J. Math. Phys. 53, 042105 (2012)10.1063/1.4705268; S. S. Bayin, J. Math. Phys. 53, 084101 (2012)]10.1063/1.4739758. We give additional details about how these integrals are evaluated and show that there is no inconsistency for an infinite square well.
Recently, it was pointed out that the solutions found in the literature for the space fractional Schrödinger equation in a piecewise manner are wrong, except the case with the delta potential. We re-analyze this problem and show that an exact and a proper treatment of the relevant integral prove otherwise. We also discuss effective potential approach and present a free particle solution for the space and time fractional Schrödinger equation in general coordinates in terms of Fox's H-functions.
In this paper, the fractional Schrödinger equation that contains the quantum Riesz fractional derivative instead of the Laplace operator is revisited for the case of a particle moving in the infinite potential well. In the recent papers [M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Schwarz, “On the nonlocality of the fractional Schrödinger equation,” J. Math. Phys. 51, 062102 (2010)10.1063/1.3430552] and [S. S. Bayin, “On the consistency of the solutions of the space fractional Schrödinger equation,” J. Math. Phys. 53, 042105 (2012)10.1063/1.4705268] published in this journal, controversial opinions regarding solutions to the fractional Schrödinger equation for a particle moving in the infinite potential well that were derived by Laskin [“Fractals and quantum mechanics,” Chaos 10, 780–790 (2000)10.1063/1.1050284] have been given. In this paper, a thorough mathematical treatment of these matters is provided. The problem under consideration is reformulated in terms of three integral equations with the power kernels. Even if the equations look not very complicated, no solution to these equations in explicit form is known. Still, the obtained equations are used to show that the eigenvalues and eigenfunctions of the fractional Schrödinger equation for a particle moving in the infinite potential well given by Laskin [“Fractals and quantum mechanics,” Chaos 10, 780–790 (2000)10.1063/1.1050284] and many other papers by different authors cannot be valid as has been first stated by Jeng et al. [“On the nonlocality of the fractional Schrödinger equation,” J. Math. Phys. 51, 062102 (2010)10.1063/1.3430552].