Semiclassical resolvent estimates in chaotic scattering

We prove resolvent estimates for semiclassical operators such as − h 2 Δ + V(x) in scattering situations. Provided the set of trapped classical trajectories supports a chaotic flow and is sufficiently filamentary, the analytic continuation of the resolvent is bounded by h − M in a strip whose width is determined by a certain topological pressure associated with the classical flow. This polynomial estimate has applications to local smoothing in Schrodinger propagation and to energy decay of solutions to wave equations.