Spectral deviations for the damped wave equation

We prove a Weyl-type fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold. It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the number of eigenvalues in a given horizontal strip deviating from this typical behaviour; the exponent that appears naturally is the `entropy' that gives the deviation rate from the Birkhoff ergodic theorem for the geodesic flow. A Weyl-type lower bound is still far from reach; but in the particular case of arithmetic surfaces, and for a strong enough damping, we can use the trace formula to prove a result going in this direction

Chaotic vibrations and strong scars

This article aims at popularizing some aspects of "quantum chaos", in particular the study of eigenmodes of classically chaotic systems, in the semiclassical (or high frequency) limit

Dispersion and controllability for the Schr\"odinger equation on negatively curved manifolds

We study the time-dependent Schr\"odinger equation $\imath\frac{\partial u}{\partial t}=-1/2\Delta u,$ on a compact riemannian manifold on which the geodesic flow has the Anosov property. Using the notion of semiclassical measures, we prove various results related to the dispersive properties of the Schr\"odinger propagator, and to controllability