On the nonlocality of the fractional Schrödinger equation

Abstract

A number of papers over the past eight years have claimed to solve the fractional Schrödinger 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 Schrödinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported ground state, which is based on a piecewise approach, is definitely not a solution of the fractional Schrödinger equation for the general fractional parameter α. On a more positive note, we present a solution to the fractional Schrödinger equation for the one-dimensional harmonic oscillator with α=1.