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Fractional Schrödinger equation for a particle moving in a potential well. (English) Zbl 1280.81044
Summary: 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, ibid. 51, No. 6, 062102, 6 p. (2010; doi:10.1063/1.3430552)] and [S. S. Bayin, ibid. 53, No. 4, 042105, 9 p. (2012; Zbl 1275.81026)] 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 N. Laskin [Chaos 10, No. 4, 780–790 (2000; Zbl 1071.81513)] 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 [loc. cit] and many other papers by different authors cannot be valid as has been first stated by Jeng et al. [loc. cit.].
©2013 American Institute of Physics

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
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