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Recent progress on the quantum unique ergodicity conjecture

Author: Peter Sarnak
Journal: Bull. Amer. Math. Soc. 48 (2011), 211-228
MSC (2010): Primary 11Fxx, 11Mxx, 35Qxx, 37Axx, 81Sxx
Published electronically: January 10, 2011
MathSciNet review: 2774090
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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

Received by editor(s): February 8, 2010
Published electronically: January 10, 2011
Article copyright: © Copyright 2011 American Mathematical Society
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