The dynamics of the Schr\"odinger flow from the point of view of semiclassical measures

TL;DR

This paper investigates the high-frequency behavior and properties of the Schr"odinger flow on compact Riemannian manifolds using semiclassical measures, highlighting differences based on curvature.

Contribution

It provides a survey of results connecting the Schr"odinger flow's dynamics with classical geodesic flow, emphasizing curvature effects and quantum-classical correspondence.

Findings

High-frequency behavior varies with curvature.

Semiclassical measures reveal regularizing properties.

Differences in dynamics between positive, negative, and zero curvature.

Abstract

On a compact Riemannian manifold, we study the various dynamical properties of the Schr\"odinger flow $(e^{it\Delta/2})$, 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.