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Analytic smoothing effect for the Schrödinger equation with long‐range perturbation

We study the microlocal analytic singularity of solutions to the Schrödinger equation with analytic coefficients. Using microlocal weight estimates developed for estimating phase space tunneling, we prove microlocal smoothing estimates that generalize results by Robbiano and 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 Schrödinger operator is a long‐range‐type perturbation of the Laplacian, and we employ positive commutator‐type estimates to prove the smoothing property. © 2005 Wiley Periodicals, Inc.