On the rate of quantum ergodicity I: Upper bounds

One problem in quantum ergodicity is to estimate the rate of decay of the sums {Mathematical expression} on a compact Riemannian manifold (M, g) with ergodic geodesic flow. Here, {λ_j, φ{symbol}_j} are the spectral data of the Δ of (M, g), A is a 0-th order ψDO, {Mathematical expression} is the (Liouville) average of its principal symbol and {Mathematical expression}. That S_k(λ;A)=o(1) is proved in [S, Z.1, CV.1]. Our purpose here is to show that S_k(λ;A)=O((log λ)^-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]).