Abstract
For a class of quantized open chaotic systems satisfying a natural dynamical assumption we show that the study of the resolvent, and hence of scattering and resonances, can be reduced to the study of a family of open quantum maps, that is of finite dimensional operators obtained by quantizing the Poincaré map associated with the flow near the set of trapped trajectories.
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Acknowledgements
We would like to thank the National Science Foundation for partial support under the grant DMS-0654436. This article was completed while the first author was visiting the Institute of Advanced Study in Princeton, supported by the National Science Foundation under agreement No. DMS-0635607. The first and second authors were also partially supported by the Agence Nationale de la Recherche under the grant ANR -09-JCJC-0099-01. Thanks also to Edward Ott for his permission to include Fig. 3 in our paper.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Nonnenmacher, S., Sjöstrand, J. & Zworski, M. From Open Quantum Systems to Open Quantum Maps. Commun. Math. Phys. 304, 1–48 (2011). https://doi.org/10.1007/s00220-011-1214-0
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DOI: https://doi.org/10.1007/s00220-011-1214-0