Abstract

We are interested in the dynamics of linear Schrödinger equations: i\partial_t u(t,x)+\Delta_x u(t,x)-V(t,x)u(t,x)=0,\quad (t,x)\in \mathbb{R}\times M, in bounded geometries such as a compact manifold, equipped with a Riemannian metric, or a bounded domain in Euclidean space. More specifically, we would like to understand the structure of those subsets on which high-frequency solutions can concentrate (in the sense of the L2 norm); that is, regions on which the position probability densities |u_n(t,x)|^2 of a normalized sequence of solutions can accumulate. This problem is also related to quantifying dispersion and understanding controllability properties for Schrödinger equations.

We give a detailed answer to this question for systems whose underlying classical dynamics (the geodesic flow or the billiard flow) is completely integrable (as flat tori, spheres or the planar disk). Our analysis is based on understanding the structure of Wigner measures associated to sequences of solutions. We accomplish that by analysing the solutions to the corresponding Wigner equations by means of a (second-micro)localization with respect to a partition of phase-space adapted to the classical dynamical system.

This talk is based on joint works with Nalini Anantharaman, Clotilde Fermanian-Kammerer, Matthieu Léautaud and Gabriel Rivière.