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Quantum decay rates in chaotic scattering

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Acta Mathematica

Abstract

We study quantum scattering on manifolds equivalent to the Euclidean space near infinity, in the semiclassical regime. We assume that the corresponding classical flow admits a non-trivial trapped set, and that the dynamics on this set is of Axiom A type (uniformly hyperbolic). We are interested in the distribution of quantum resonances near the real axis. In two dimensions, we prove that, if the trapped set is sufficiently “thin”, then there exists a gap between the resonances and the real axis (that is, quantum decay rates are bounded from below). In higher dimension, the condition for this gap is given in terms of a certain topological pressure associated with the classical flow. Under the same assumption, we also prove a resolvent estimate with a logarithmic loss compared to non-trapping situations.

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Authors and Affiliations

  1. Institut de Physique Théorique, CNRS URA 2306, CEA-Saclay, FR-91191, Gif-sur-Yvette, France

    Stéphane Nonnenmacher

  2. University of California, Berkeley, Department of Mathematics, Berkeley, CA, 94720, U.S.A.

    Maciej Zworski

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  1. Stéphane Nonnenmacher
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  2. Maciej Zworski
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Correspondence to Stéphane Nonnenmacher.

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Nonnenmacher, S., Zworski, M. Quantum decay rates in chaotic scattering. Acta Math 203, 149–233 (2009). https://doi.org/10.1007/s11511-009-0041-z

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  • DOI: https://doi.org/10.1007/s11511-009-0041-z

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