Semiclassical measures and the Schroedinger flow on Riemannian manifolds

TL;DR

This paper investigates the behavior of semiclassical measures for solutions to the Schrödinger equation on Riemannian manifolds, providing characterizations for Euclidean space and Zoll manifolds, and exploring phenomena like resonances and non-classical behavior.

Contribution

It offers a comprehensive analysis of semiclassical measures under long-time evolution, including a weak Egorov theorem and new insights into resonant effects on flat tori.

Findings

Complete characterization of semiclassical measures on Euclidean space and Zoll manifolds.

Identification of non-classical behavior due to resonances on flat tori.

Example showing evolution measures are not uniquely determined by initial measures.

Abstract

In this article we study limits of Wigner distributions (the so-called semiclassical measures) corresponding to sequences of solutions to the semiclassical Schroedinger equation at times scales $\alpha_{h}$ tending to infinity as the semiclassical parameter $h$ tends to zero (when $\alpha _{h}=1/h$ this is equivalent to consider solutions to the non-semiclassical Schreodinger equation). Some general results are presented, among which a weak version of Egorov's theorem that holds in this setting. A complete characterization is given for the Euclidean space and Zoll manifolds (that is, manifolds with periodic geodesic flow) via averaging formulae relating the semiclassical measures corresponding to the evolution to those of the initial states. The case of the flat torus is also addressed; it is shown that non-classical behavior may occur when energy concentrates on resonant frequencies. Moreover, we present an example showing that the semiclassical measures associated to a sequence of states no longer determines those of their evolutions. Finally, some results concerning the equation with a potential are presented.