Analytic Wave Front Set for Solutions to Schrödinger Equations II – Long Range Perturbations

This paper is a continuation of a previous work, where short range perturbations of the flat Euclidian metric where considered. Here, we generalize the results to long-range perturbations (in particular, we can allow potentials subquadratic at infinity). More precisely, we construct a modified quantum free evolution acting on Sjöstrand’s spaces, and we characterize the analytic wave front set of the solution to the Schrödinger equation, in terms of the microlocal semiclassical exponential decay of the corresponding modified quantum free evolution. The result is valid for t < 0 near the forward non trapping points, and for t > 0 near the backward non trapping points.