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Abstract. 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 (very chaotic). The final objective is to obtain conditions (in terms of the geodesic flow on X, the structure of the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. The spectrum of the equation amounts to a nonselfadjoint spectral problem. Using semiclassical methods, we derive estimates and upper bounds for the high frequency spectral distribution, in terms of dynamically defined quantities, like the value distribution of the time-averaged damping. We also consider the toy model of damped quantized chaotic maps, for which we derive similar estimates, as well as a new upper bound for the spectral radius depending on the set of minimally damped trajectories. Contents

Abstract. We determine the general form of the asymptotics for Dirichlet eigenvalues of the one–dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion. We also derive a trace formula for this problem. 1.

geometrical quantity in damped wave equations on a square