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

TL;DR

This paper addresses the controversy over the consistency of solutions to the space fractional Schrödinger equation, demonstrating that the integral representation in momentum space is consistent and clarifying the role of the Riesz derivative.

Contribution

The paper clarifies the consistency of the infinite square well solutions and analyzes the Riesz derivative representations to resolve previous disagreements.

Findings

Integral representation in momentum space is consistent for general n.

Different Riesz derivative representations have the same Fourier transform under consistent assumptions.

The controversy over solution consistency is resolved with these clarifications.

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