Spectral theory of damped quantum chaotic systems

Details Spectral theory of damped quantum chaotic systems Spectral theory of damped quantum chaotic systems

2011-09-05

arxiv.org/abs/1109.0930v1

Physics

Mathematical Physics

Lecture given at the Journ\'ees semiclassiques 2011, Biarritz, 6-10 June 2011

Scientific paper

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.

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