On the nonlocality of the fractional Schr\"{o}dinger equation

TL;DR

This paper critiques previous solutions to the fractional Schr"{o}dinger equation, emphasizing its inherent nonlocality, and demonstrates that certain piecewise solutions are invalid, while providing a valid solution for the harmonic oscillator at a specific fractional parameter.

Contribution

It clarifies the nonlocal nature of the fractional Schr"{o}dinger equation and corrects misconceptions about piecewise solutions, offering a valid solution for the harmonic oscillator.

Findings

Piecewise solutions for the infinite square well are invalid for general fractional parameters.

The fractional Schr"{o}dinger equation's nonlocality prevents standard piecewise solutions.

A valid solution is provided for the harmonic oscillator at =1.

Abstract

A number of papers over the past eight years have claimed to solve the fractional Schr\"{o}dinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schr\"{o}dinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported groundstate, which is based on a piecewise approach, is definitely not a solution of the fractional Schr\"{o}dinger equation for general fractional parameters $\alpha$. On a more positive note, we present a solution to the fractional Schr\"{o}dinger equation for the one-dimensional harmonic oscillator with $\alpha=1$.