  This paper is devoted to the analysis of propagation properties for the
solutions of a one-dimensional non-local Schr\"odinger equation involving the
fractional Laplace operator $(-d_x^2)^s$, $s\in(0,1)$. We adopt a classical WKB
approach and we provide a systematic procedure for building a suitable ansatz
for the solutions to the problem. In this way, we can obtain quasi-solutions
which are localized along the rays of geometric optics, whose group velocity
can be computed explicitly in terms of the parameter $s$. Our results are then
confirmed by numerical simulations, based on a finite element approximation of
the fractional Laplacian and on a Crank-Nicholson scheme for the time
integration. As an application, the controllability problem for the fractional
Schr\"odinger equation is analyzed, finding confirmations of previously known
results.

Analysis of PDEs