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

WKB expansion for a fractional Schr\"odinger equation with applications to controllability