Consistency Problem of the Solutions of the Space Fractional Schr\"odinger Equation

Abstract

Recently, consistency of the infinite square well solution of the space fractional Schr\"odinger equation has been the subject of some controversy. In [J. Math. Phys. 54, 014101 (2013)], Hawkins and Schwarz objected to the way certain integrals are evaluated to show the consistency of the infinite square well solutions of the space fractional Schr\"odinger equation [J. Math. Phys. 53, 042105 (2012); J. Math. Phys. 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. PACS numbers: 03.65.Ca, 02.50.Ey, 02.30.Gp, 03.65.Db