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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)

- NILS DENCKER, JOHANNES SJÖSTRAND, MACIEJ ZWORSKI
- 2003

- 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)

Any compact C ∞ manifold with boundary admits a Riemann metric on its interior taking the form x −4 dx 2 + x −2 h 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… (More)

The purpose of this article is to discuss a simple linear algebraic tool which has proved itself very useful in the mathematical study of spectral problems arising in elecromagnetism and quantum mechanics. Roughly speaking it amounts to replacing an operator of interest by a suitably chosen invertible system of operators. That approach has a very long… (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)

- STÉPHANE NONNENMACHER, MACIEJ ZWORSKI, M. ZWORSKI
- 2007

In this article we prove that for a large class of operators, including Schrödinger operators, 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.… (More)

Quantum ergodicity of classically chaotic systems has been studied extensively both theoretically and experimentally, in mathematics, and in physics. Despite this long tradition we are able to present a new rigorous result using only elementary calculus. In the case of the famous Bunimovich billiard table shown in Fig.1 we prove that the wave functions have… (More)

- Laurent Guillop, Maciej Zworski
- 2007

We present a wave group version of the Selberg trace formula for an arbitrary surface of nite geometry. As an application we give a new lower bound on the number of resonances for hyperbolic surfaces. Motivated by recent results we formulate a conjecture on a lower bound for the counting function of resonances in a strip.

This letter summarizes numerical results from [1] which show that in quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy E scales like ¯ h − D(K E)+1 2. Here, K E denotes the subset of the classical energy surface {H = E} which stays bounded for all time under the flow of H and D (K E) denotes its fractal… (More)