The dynamics of the Schrödinger flow from the point of view of semiclassical measures

Abstract

On a compact Riemannian manifold, we study the various dynamical properties of the Schrödinger flow $(e^{it\Delta /2})$, through the notion of semiclassical measures and the quantum-classical correspondence between the Schrödinger equation and the geodesic flow. More precisely, we are interested in its high-frequency behavior, as well as its regularizing and unique continuation-type properties. We survey a variety of results illustrating the difference between positive, negative and vanishing curvature.

arXiv commentSurvey article; 24 pages. References and a Section 1.4 added