Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Abstract

The Quantum Unique Ergodicity (QUE) conjecture of Rudnick-Sarnak is that every eigenfunction phi_n of the Laplacian on a manifold with uniformly-hyperbolic geodesic flow becomes equidistributed in the semiclassical limit (eigenvalue E_n -> infinity), 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 <phi_n, A phi_m> of a piecewise-constant test function A. By collecting 30000 diagonal elements (up to level n ~ 7*10^5) we find that their variance decays with eigenvalue as a power 0.48 +- 0.01, close to the estimate 1/2 of Feingold-Peres (FP). This contrasts the results of existing studies, which have been limited to E_n a factor 10^2 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 FP at the highest accuracy (0.7%) thus far in any chaotic system. We outline the efficient scaling method used to calculate eigenfunctions.