High-frequency resolvent estimates on asymptotically Euclidean warped products

TL;DR

This paper investigates high-frequency resolvent estimates on asymptotically Euclidean warped product manifolds, classifying their behavior based on trapping stability, and applies results to resonance free regions and billiard dynamics.

Contribution

It introduces a microlocal analysis framework for resolvent behavior on warped products, linking trapping stability to resolvent growth and resonance distribution.

Findings

Weakly stable trapping allows highly concentrated quasimodes and fast resolvent growth.

Weakly unstable trapping enforces spreading of quasimodes.

Either a resonance free region exists or resonances approach the real axis faster than polynomial rate.

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 that the high-frequency resolvent is either bounded by $C_\epsilon |\lambda|^\epsilon$ for any $\epsilon>0$, 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 Schr\"odinger operator. We then classify the microlocal resolvent behaviour 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. As a first application, we conclude that either there is a resonance free region of size $| \Im \lambda | \leq C_\epsilon | \Re \lambda |^{-1-\epsilon}$ for any $\epsilon>0$, or there is a sequence of resonances converging to the real axis faster than any polynomial. Again, a stronger result holds if the manifold is analytic. As a second application, we prove a spreading result for weak quasimodes in partially rectangular billiards.