Recent progress on the quantum unique ergodicity conjecture

By Peter Sarnak. 2011

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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 micro-local 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. The recent account by Zelditch [Zel4] covers some of the same material and his and our discussion complement each other nicely, as he goes into more detail with the first category and we with the second. 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. Recent Progress on QUE 2 §1. The General QUE Conjecture In Figure 1 the domains ΩE,ΩS and ΩB are an ellipse, a stadium and a “Barnett” billiard table respectively. Superimposed on these are the densities of a consecutive sequence of high frequency eigenfunctions (“states”, “modes”) of the Laplacian. That is they are solutions to  4φj + λjφj = 0 in Ω . φ ∣∣ ∂Ω = 0 (Dirichlet boundary conditions), ∫ Ω |φj| dxdy = 1 . (0) Here 4 = divgrad = ∂2 ∂x2 + ∂ 2 ∂y2 , λ1 < λ2 ≤ λ3 . . . are the eigenvalues and the eigenfunctions are normalized to have unit L-norm. The sequences are of 12 consecutive modes around the 5600 eigenvalue. They are ordered from left to right and then down and the grayscale represents the probability density |φ|, with zero white and larger values darker.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/fdc3c46c6a14aa67947dfd3ea29a3f0ff3d32a26

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Citations in the corpus

Geometrical Structure of Laplacian Eigenfunctions (2012, background): “In particular, some orbits of a particle moving in a classical billiard may be reflected as “scars” in the spatial structure of eigenfunctions in the related quantum billiard [22, 53, 173, 174, 182, 183, 199, 210, 212, 351, 352, 372, 373].”
Real zeros of holomorphic Hecke cusp forms (2011, background): “ak One of the striking consequences of the recent proof of the holomorphic QUE conjecture by Holowinsky and Soundararajan [HS] is that Z(f) is equidistributed in X as k!1(see also [Ru] and the report [Sa-2] for a discussion). That is for any nice set ˆX jZ(f) \ j jZ(f)j ! Area( ) Area(X) ; (3) as k!”