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Delocalization of Slowly Damped Eigenmodes on Anosov Manifolds

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Abstract

We look at the properties of high frequency eigenmodes for the damped wave equation on a compact manifold with an Anosov geodesic flow. We study eigenmodes with damping parameters which are asymptotically close enough to the real axis. We prove that such modes cannot be completely localized on subsets satisfying a condition of negative topological pressure. As an application, one can deduce the existence of a “strip” of logarithmic size without eigenvalues below the real axis under this dynamical assumption on the set of undamped trajectories.

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Authors and Affiliations

  1. Laboratoire Paul Painlevé (U.M.R. CNRS 8524), U.F.R. de Mathématiques, Université Lille 1, 59655, Villeneuve d’Ascq Cedex, France

    Gabriel Rivière

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  1. Gabriel Rivière
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Correspondence to Gabriel Rivière.

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Communicated by S. Zelditch

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Rivière, G. Delocalization of Slowly Damped Eigenmodes on Anosov Manifolds. Commun. Math. Phys. 316, 555–593 (2012). https://doi.org/10.1007/s00220-012-1588-7

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