High-Frequency Dynamics for the Schrödinger Equation, with Applications to Dispersion and Observability

Abstract

In these notes we review various aspects of the high-frequency dynamics of solutions to the linear Schrodinger equation. Special attention is paid to the influence on the dynamics of the underlying geometry and the perturbation terms (as potentials, for instance). We treat in detail the by now classical results on the semiclassical limit: first by describing the WKB asymptotic expansion and further by presenting a complete description in terms of semiclassical/Wigner measures. We then address the main theme of these notes: the use of tools from the analysis of the semiclassical limit (such as Wigner measures) to obtain a description of the high-frequency structure of the solutions to the non-semiclassical Schrodinger equation (i.e. where no semiclassical small parameter is involved). This issue has a number of connections with other dynamical properties of the equation that have been extensively studied in the literature, such as dispersive effects, Strichartz estimates and unique continuation-type properties (which are relevant in control theory and inverse problems). A complete description is given in the case in which the underlying geometry is a manifold with periodic geodesic flow (Zoll manifolds) and for the torus, where we present the main ideas of the recent work of the author in collaboration with Anantharaman and Macia (J. Eur. Math. Soc. (JEMS), 16(6), 1253–1288, 2014).