Abstract. For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, P(h) = −h 2 ∆g + V (x), on compact Riemannian manifolds, we give logarithmic lower bounds on the mass of eigenfunctions outside neighbourhoods of generic closed hyperbolic orbits. More precisely we show that if A is a pseudodifferential operator which is microlocally equal to the identity near the hyperbolic orbit and microlocally zero away from the orbit, then ‖u ‖ ≤ C ( √ log(1/h)/h)‖P(h)u ‖ + C √ log(1/h)‖(I − A)u ‖. This generalizes earlier estimates of Colin de Verdière-Parisse [CVP] obtained for a special case, and of Burq-Zworski [BuZw] for real hyperbolic orbits. 1.

Abstract. For a large class of complete, non-compact Riemannian manifolds, (M, g), with boundary, we prove high energy resolvent estimates in the case where there is one trapped hyperbolic geodesic. As an application, we have the following local smoothing estimate for the Schrödinger propagator:

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Abstract. We compute the waves propagating on the compact surface of constant negative curvature and genus 2. We adopt a variational approach using finite elements. We have to implement the action of the fuchsian group by suitable boundary conditions of periodic type. A spectral analysis of the wave allows to compute the spectrum and the eigenfunctions of the Laplace-Beltrami operator. We test the exponential decay due to a localized dumping and the ergodicity of the geodesic flow.

These lectures originate with a course MZ taught at UC Berkeley during the spring semester of 2003, notes for which LCE took in class. In this presentation we have tried hard to work out the full details for many proofs only sketched in class. In addition, we have reworked the order of presentation, added additional topics, and included more heuristic commentary. We have as well introduced consistent notation, much of which is collected into Appendix A. Relevant functional analysis and other facts have been consolidated into Appendices B–D. We should mention that two excellent treatments of mathematical semiclassical analysis have appeared recently. The book by Dimassi and Sjöstrand [D-S] starts with the WKB-method, develops the general semiclassical calculus, and then provides “high tech ” spectral asymptotics. The presentation of Martinez [M] is based on a systematic development of FBI (Fourier-Bros-Iagolnitzer) transform techniques, with applications to microlocal exponential estimates and propagation estimates. These notes are intended as a more elementary and broader introduction. Except for the general symbol calculus, where we followed Chapter 7 of [D-S], there is little overlap with these other two texts, or with the early and influential book of Robert [R]. In his study of semiclassical calculus MZ has been primarily influenced by his long collaboration with Johannes Sjöstrand and he acknowledges that with pleasure and gratitude. Our thanks to Faye Yeager for typing a first draft and to Hans Christianson for his careful reading of earlier versions of these notes. And thanks also to Jonathan Dorfman for TeX advice. We are quite aware that many errors remain in our exposition, and so we ask our readers to please send any comments or corrections to us at

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

geometrical quantity in damped wave equations on a square