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

TL;DR

This paper reexamines solutions to the space fractional Schrödinger equation, demonstrating that previous piecewise solutions were incorrect except for the delta potential case, and provides exact solutions using advanced integral methods and Fox's H-functions.

Contribution

It offers a rigorous integral-based analysis that corrects prior misconceptions and presents new exact solutions for the fractional Schrödinger equation in various coordinate systems.

Findings

Previous piecewise solutions are generally incorrect.

An exact integral approach yields correct solutions.

New solutions expressed in terms of Fox's H-functions.

Abstract

Recently it was pointed out that the solutions found in literature for the space fractional Schr\"odinger equation in a piecewise manner are wrong, except the case with the delta potential. We reanalyze this problem and show that an exact and a proper treatment of the relevant integral proves otherwise. We also discuss effective potential approach and present a free particle solution for the space and time fractional Schr\"odinger equation in general coordinates in terms of Fox's H-functions.