Fractal Weyl law for open quantum chaotic maps

Abstract

We study a semiclassical quantization of Poincaré maps arising in scattering problems with fractal hyperbolic trapped sets. The main application is the proof of a fractal Weyl upper bound for the number of resonances/scattering poles in small domains near the real axis. This result includes the case of several convex (hard) obstacles satisfying a no-eclipse condition.

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    @book {EZB, MRKEY = {2952218},
      AUTHOR = {Zworski, Maciej},
      TITLE = {Semiclassical Analysis},
      SERIES = {Grad. Studies in Math.},
      VOLUME = {138},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2012},
      PAGES = {xii+431},
      ISBN = {978-0-8218-8320-4},
      MRCLASS = {58J40 (35P05 35S05 81Q20)},
      MRNUMBER = {2952218},
      MRREVIEWER = {David Borthwick},
      ZBLNUMBER = {1252.58001},
      }

Authors

Stéphane Nonnenmacher

Institut de Physique Théorique, CEA/DSM/PhT, Unité de recherche associée au CNRS, CEA-Saclay, Gif-sur-Yvette, France

Johannes Sjöstrand

Institut de Mathématiques de Bourgogne, Université de Bourgogne, UFR Science et Techniques, Dijon Cedex, France

Maciej Zworski

University of California at Berkeley, Berkeley, CA