The dynamics of the Schr\"odinger flow from the point of view of semiclassical measures
Abstract
On a compact Riemannian manifold, we study the various dynamical properties of the Schr\"odinger flow (eit/2), through the notion of semiclassical measures and the quantum-classical correspondence between the Schr\"odinger 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.
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