WKB expansion for a fractional Schrödinger equation with applications to controllability

Abstract

This paper is devoted to the analysis of propagation properties for the solutions of a one-dimensional non-local Schrödinger 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 Schrödinger equation is analyzed, finding confirmations of previously known results.