On the rate of quantum ergodicity I: Upper bounds

Abstract

One problem in quantum ergodicity is to estimate the rate of decay of the sums

$$S_k (\lambda ;A) = \frac{1}{{N(\lambda )}}\sum\limits_{\sqrt {\lambda _j } \leqq \lambda } {\left| {(A\varphi _j ,\varphi _j ) - \bar \sigma _A } \right|^k } $$

on a compact Riemannian manifold (M, g) with ergodic geodesic flow. Here, {λ _j ,ϕ _j } are the spectral data of the Δ of(M, g), A is a 0-th order ψDO,$\bar \sigma _A$ is the (Liouville) average of its principal symbol and$N(\lambda ) = \# \{ j:\sqrt {\lambda _j } \leqq \lambda \}$. ThatS _k (λ;A)=o(1) is proved in [S, Z.1, CV.1]. Our purpose here is to show thatS _k (λ;A)=O((logλ)^−k/2) on a manifold of (possibly variable) negative curvature. The main new ingredient is the central limit theorem for geodesic flows on such spaces ([R, Si]).