The Schrödinger Flow in a Compact Manifold: High-frequency Dynamics and Dispersion

Abstract

We discuss various aspects of the dynamics of the Schrodinger flow on a compact Riemannian manifold that are related to the behavior of highfrequency solutions. In particular we show that dispersive (Strichartz) estimates fail on manifolds whose geodesic flow is periodic (thus generalizing a well-known result for spheres proved via zonal spherical harmonics). We also address the issue of the validity of observability estimates. We show that the geometric control condition is necessary in manifolds with periodic geodesic flow and we give a new, geometric, proof of a result of Jaffard on the observability for the Schrodinger flow on the two-torus. All our proofs are based on the study of the structure of semiclassical (Wigner) measures corresponding to solutions to the Schrodinger equation.