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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)
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)
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)
We present a conjecture relating the density of quantum resonances for an open chaotic system to the fractal dimension of the associated classical repeller. Mathematical arguments justifying this conjecture are discussed. Numerical evidence based on computation of resonances of systems of n disks on a plane are presented supporting this conjecture. The(More)
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)