Fractional Schrödinger equation for a particle moving in a potential well.

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)] and [S. S. Bayin, 'On the consistency of the solutions of the space fractional Schrödinger equation,' J. Math. Phys. 53, 042105 (2012)] 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)] 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)] 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)]. [ABSTRACT FROM AUTHOR]