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DELOCALIZATION OF SLOWLY DAMPED EIGENMODES ON ANOSOV

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DELOCALIZATION OF SLOWLY DAMPED EIGENMODES ON ANOSOV MANIFOLDS GABRIEL RIVIÈRE Abstract. We look at the properties of high frequency eigenmodes for the damped wave equa- tion 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 eigenval- ues below the real axis under this dynamical assumption on the set of undamped trajectories. 1. Introduction Let M be a smooth, compact, connected Riemannian manifold of dimension d ≥ 2 and without boundary. We will be interested in the high frequency analysis of the damped wave equation, ( ∂2t ?∆ + 2V (x)∂t ) u(x, t) = 0, u(x, 0) = u0, ∂tu(x, 0) = u1, where ∆ is the Laplace-Beltrami operator on M and V ? C∞(M,R+) is the damping function. This problem can be rewritten as (1) (?ı∂t +A)u(t) = 0, where u(t) := (u(t), ı∂tu(t)) and (2) A = ( 0 Id ?∆ ?2ıV ) .

  • gt

  • gt-invariant subset

  • eigenmodes

  • geometric control

  • probability µ

  • semiclassical measures

  • see also

  • call semiclassical

  • frequency limit


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Informations

Publié par
Nombre de lectures 65

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