Analytic Singularities for Long Range Schr"odinger Equations

We consider the Schrödinger equation associated to long range perturbations of the flat Euclidean metric (in particular, potentials growing subquadratically at infinity are allowed). We construct a modified quantum free evolution G(s) acting on Sjöstrand’s spaces, and we characterize the analytic wave front set of the solution e?itH u_0 of the Schrödinger equation, in terms of the semiclassical exponential decay of G(?th^{?1})Tu_0, where T stands for the Bargmann-transform. The result is valid for t < 0 near the forward non-trapping points, and for t > 0 near the backward non-trapping points.