Title: Semiclassical evolution of Lagrangian states into random waves.

Abstract: In 1977, Berry conjectured that eigenfunctions of the Laplacian on manifolds of negative curvature behave, in the high-energy (or semiclassical) limit, as a random superposition of plane waves. This conjecture, central in quantum chaos, is still completely open. In this talk, we will consider a much simpler situation. On a manifold of negative curvature, we will consider a Lagrangian state (the generalization of a plane wave on a manifold). We aim to show that, when evolved during a long time by the Schrödinger equation, these functions do behave, in the semiclassical limit, as a random superposition of plane waves. We will show such a result in two situations: when the phase of the Lagrangian state is generic, and when the Laplace-Beltrami operator is perturbed by a small generic potential.

This is based on joint work with Alejandro Rivera, and with Martin Vogel.