Semiclassical measures and the Schrödinger flow on Riemannian manifolds

Abstract

In this paper we study limits of Wigner distributions (the so-called semiclassical measures) corresponding to sequences of solutions to the semiclassical Schrödinger equation at time scales αh tending to infinity as the semiclassical parameter h tends to zero (when αh = 1/h this is equivalent to considering solutions to the non-semiclassical Schrödinger equation). Some general results are presented, among which is 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 behaviour may occur when energy concentrates on resonant frequencies. Moreover, we present an example showing that the semiclassical measures associated with a sequence of states no longer determines those of their evolutions. Finally, some results concerning the equation with a potential are presented.