Analytic smoothing effect for the Schroedinger equation with long range perturbation

We study microlocal analytic singularity of solutions to Schr"odinger equation with analytic coefficients. Using microlocal weight estimate developped for estimating the phase space tunneling, we prove microlocal smoothing estimates that generalize results by L. Robbiano and C. Zuily. We show that the exponential decay of the initial state in a cone in the phase space implies microlocal analytic regularity of the solution at a positive time. We suppose the Schr"odinger operator is a long-range type perturbation of the Laplacian, and we employ positive commutator type estimates to prove the smoothing property.