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1. Statement of the results This paper describes the connection between scattering matrices on conformally compact asymptotically Einstein manifolds and conformally invariant objects on their boundaries at infinity. This connection is a manifestation of the general principle that the far field phenomena on a conformally compact Einstein manifold are related… (More)

Any compact C∞ manifold with boundary admits a Riemann metric on its interior taking the form x−4dx2 + x−2h′ near the boundary, where x is a boundary defining function and h′ is a smooth symmetric 2-cotensor restricting to be positive-definite, and hence a metric, h, on the boundary. The scattering theory associated to the Laplacian for such a ‘scattering… (More)

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)

- NILS DENCKER, Johannes Sjoestrand, Maciej Zworski
- 2003

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)

- WAVESSIU - HUNG TANG, Maciej Zworski
- 2007

- Maciej Zworski
- 2007

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)

The purpose of this paper is to show how ideas coming from scattering theory (resolvent estimates) lead to results in control theory and to some closely related eigenfunction estimates. The black box approach in scattering theory developed by Sjöstrand and the second author [37] puts scattering problems with different structures in one framework and allows… (More)

We prove that for evolution problems with normally hyperbolic trapping in phase space, correlations decay exponentially in time. Normally hyperbolic trapping means that the trapped set is smooth and symplectic and that the flow is hyperbolic in directions transversal to it. Flows with this structure include contact Anosov flows [23],[46],[47], classical… (More)