Consistency problem of the solutions of the space fractional Schrödinger equation

Recently, consistency of the infinite square well solution of the space fractional Schrodinger equation has been the subject of some controversy. Hawkins and Schwarz [J. Math. Phys. 54, 014101 (2013)] objected to the way certain integrals are evaluated to show the consistency of the infinite square well solutions of the space fractional Schrodinger equation [S. S. Bayin, J. Math. Phys. 53, 042105 (2012); 53, 084101 (2012)]. Here, we show for general n that as far as the integral representation of the solution in the momentum space is concerned, there is no inconsistency. To pinpoint the source of a possible inconsistency, we also scrutinize the different representations of the Riesz derivative that plays a central role in this controversy and show that they all have the same Fourier transform, when evaluated with consistent assumptions. (C) 2013 AIP Publishing LLC

Definition of the Riesz derivative and its application to space fractional quantum mechanics

We investigate and compare different representations of the Riesz derivative, which plays an important role in anomalous diffusion and space fractional quantum mechanics. In particular, we show that a certain representation of the Riesz derivative, Rax, that is generally given as also valid for alpha = 1, behaves no differently than the other definition given in terms of its Fourier transform. In the light of this, we discuss the alpha -> 1 limit of the space fractional quantum mechanics and its consistency. Published by AIP Publishing