High-frequency resolvent estimates on asymptotically Euclidean warped products

Abstract

We consider the resolvent on asymptotically Euclidean warped product manifolds in an appropriate 0-Gevrey class, with trapped sets consisting of only finitely many components. We prove a new dichotomy: the high-frequency resolvent is either bounded by for any , or blows up faster than any polynomial (at least along a subsequence). A stronger result holds if the manifold is analytic. The method of proof is to exploit the warped product structure to separate variables, obtaining a one-dimensional semiclassical Schrödinger operator. We then classify the microlocal resolvent behavior associated to every possible type of critical value of the potential, and translate this into the associated resolvent estimates. Weakly stable trapping admits highly concentrated quasimodes and fast growth of the resolvent. Conversely, using a delicate inhomogeneous blowup procedure loosely based on the classical positive commutator argument, we show that any weakly unstable trapping forces at least some spreading of quasimodes. Several applications are given for spreading of quasimodes in partially rectangular billiards, decay rates for damped wave equations.