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

A wide variety of stochastic processes are more general than the familiar Brownian motion, but presumably can still be described by modifying the diffusion equation using a fractional Laplacian operator. In analogy with fractional diffusion, the fractional Schr\"{o}dinger equation is the ordinary Schr\"{o}dinger equation with the fractional Laplacian operator replacing the ordinary one. Over the past eight years, a number of papers 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 $\alpha=1$. Potential physical applications will also be discussed.