Journal of Modern Dynamics (JMD)

Remarks on quantum ergodicity

Pages: 119 - 133, Issue 1, March 2013      doi:10.3934/jmd.2013.7.119

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Gabriel Rivière - Laboratoire Paul Painlevé (UMR CNRS 8524), UFR de Mathématiques, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France (email)

Abstract: We prove a generalized version of the Quantum Ergodicity Theorem on smooth compact Riemannian manifolds without boundary. We apply it to prove some asymptotic properties on the distribution of typical eigenfunctions of the Laplacian in geometric situations in which the Liouville measure is not (or not known to be) ergodic.

Keywords:  Eigenfunctions of the Laplacian, quantum ergodicity, nonuniformly hyperbolic dynamical systems.
Mathematics Subject Classification:  Primary: 58J51, 35P20; Secondary: 37D25.

Received: October 2012;      Revised: January 2013;      Available Online: May 2013.