2011/09/05 by Stéphane Nonnenmacher, Nonnenmacher, Stéphane
Computer Science · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems #math-ph #math.AP #math.MP
paper · pdf · doi:10.48550/arxiv.1109.0930
Lecture given at the Journées semiclassiques 2011, Biarritz, 6-10 June 2011
arxiv created 2011/09/05 · openalex publication_date 2011/09/05 · arxiv updated 2011/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov. The main application is to obtain conditions (in terms of the geodesic flow on X and the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. We review various estimates for the high frequency spectrum in terms of dynamically defined quantities, like the value distribution of the time-averaged damping. We also present a new condition for a spectral gap, depending on the set of minimally damped trajectories.