Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

The Quantum Unique Ergodicity (QUE) conjecture of RudnickSarnak 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 30000 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 estimate 1/2 of FeingoldPeres (FP). This contrasts the results of existing studies, which have been limited to En 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. ∗now at Department of Mathematics, Dartmouth College, 6188 Bradley Hall, Hanover, NH, 03755. [email protected]