Maciej Zworski

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The purpose of this note is to apply the methods of geometric scattering theory developed by Briet-Combes-Duclos [6], Gérard-Sjöstrand [14], Mazzeo-Melrose [22] and the second author [30] in the simplest model of a Black Hole: the De SitterSchwarzschild metric. We show that the resonances (or the quasi normal modes, in the terminology of Chandrasekhar [8])(More)
We give pole free strips and estimates for resolvents of semiclassical operators which, on the level of the classical flow, have normally hyperbolic smooth trapped sets of codimension two in phase space. Such trapped sets are structurally stable – see §1.2 – and our motivation comes partly from considering the wave equation for slowly rotating Kerr black(More)
The purpose of this paper is to show how some results from the theory of partial differential equations apply to the study of pseudospectra of non-self-adjoint operators, which is a topic of current interest in applied mathematics; see [6, 29]. We will consider operators that arise from the quantization of bounded functions on the phase space T ∗Rn . For(More)
(1.1) P (h) = −h∆+ V (x) , V ∈ C c (X) , X = R 2 , with hyperbolic classical flows, the smallness of dimension of the trapped set implies that there is a gap between the resonances and the real axis. In other words, the quantum decay rate is bounded from below if the classical repeller is sufficiently filamentary. The higher dimensional statement is given(More)
The purpose of this paper is to show how the methods of Sjj ostrand for proving the geometric bounds for the density of resonances 28] apply to the case of convex co-compact hyperbolic surfaces. We prove that the exponent in the Weyl estimate for the number of resonances in subconic neighbourhoods of the continuous spectrum is related to the dimension of(More)