ON THE RATE OF QUANTUM ERGODICITY .1. UPPER-BOUNDS

One problem in quantum ergodicity is to estimate the rate of decay of the sums [GRAPHICS] on a compact Riemannian manifold (My) with ergodic geodesic flow. Here, (LAMBDA(J), OHI(J)} are the spectral data of the DELTA of (M, g), A is a 0-th order psiDO, sigma(A) is the (Liouville) average of its principal symbol and N(lambda) = # {j: square-root lambda(j) less-than-or-equal-to lambda}. That S(k)(lambda; A) = o(1) is proved in [S, Z.1, CV.1]. Our purpose here is to show that S(k)(lambda; A) = O((log lambda)-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]).