Semiclassical limits for discrete and continuous wave equations

Wigner/Semiclassical measures are a standard tool for the study of propagation of concentration and oscillation effects for solutions to wave equations. We shall illustrate this theory adressing two specific issues:

1. Discrete Wave equations on the torus. We shall give a discrete version of the theory of Wigner measures and apply it to characterize the high frequency behavior of discrete waves, recovering in particular classical results on group velocity.

2. Characterization of Semiclassical measures associated to solutions of the Schr"odinger equation on compact manifolds. In particular, we shall focus on the relation between the dynamics of the geodesic flow on a compact manifold and the structure of Wigner measures associated to the corresponding Schrödinger equation.