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Bulletin of the American Mathematical Society

Published by the American Mathematical Society, the Bulletin of the American Mathematical Society (BULL) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.47.

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Recent progress on the quantum unique ergodicity conjecture
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by Peter Sarnak PDF
Bull. Amer. Math. Soc. 48 (2011), 211-228 Request permission

Abstract:

We report on some recent striking advances on the quantum unique ergodicity, or QUE conjecture, concerning the distribution of large frequency eigenfunctions of the Laplacian on a negatively curved manifold. The account falls naturally into two categories. The first concerns the general conjecture where the tools are more or less limited to microlocal analysis and the dynamics of the geodesic flow. The second is concerned with arithmetic such manifolds where tools from number theory and ergodic theory of flows on homogeneous spaces can be combined with the general methods to resolve the basic conjecture as well as its holomorphic analogue. Our main emphasis is on the second category, especially where QUE has been proven. This note is not meant to be a survey of these topics, and the discussion is not chronological. Our aim is to expose these recent developments after introducing the necessary backround which places them in their proper context.
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Additional Information
  • Peter Sarnak
  • Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
  • MR Author ID: 154725
  • Email: sarnak@math.princeton.edu
  • Received by editor(s): February 8, 2010
  • Published electronically: January 10, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Bull. Amer. Math. Soc. 48 (2011), 211-228
  • MSC (2010): Primary 11Fxx, 11Mxx, 35Qxx, 37Axx, 81Sxx
  • DOI: https://doi.org/10.1090/S0273-0979-2011-01323-4
  • MathSciNet review: 2774090