Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Abstract

The quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak is that every eigenfunction ϕn of the Laplacian on a manifold with uniformly hyperbolic geodesic flow becomes equidistributed in the semiclassical limit (eigenvalue En ∞); that is, “strong scars” are absent. We study numerically the rate of equidistribution for a uniformly hyperbolic, Sinai-type, planar Euclidean billiard with Dirichlet boundary condition (the “drum problem”) at unprecedented high E and statistical accuracy, via the matrix elements 〈ϕn, Âϕm〉 of a piecewise-constant test function A. By collecting 30,000 diagonal elements (up to level n ≈ 7 × 105) we find that their variance decays with eigenvalue as a power 0.48 ± 0.01, close to the semiclassical estimate ½ of Feingold and Peres. This contrasts with the results of existing studies, which have been limited to En a factor 102 smaller. We find strong evidence for QUE in this system. We also compare off-diagonal variance as a function of distance from the diagonal, against Feingold-Peres (or spectral measure) at the highest accuracy (0.7%) thus far in any chaotic system. We outline the efficient scaling method and boundary integral formulae used to calculate eigenfunctions. © 2006 Wiley Periodicals, Inc.