Lower resolvent bounds and Lyapunov exponents

TL;DR

This paper establishes a polynomial lower bound on the scattering resolvent by constructing a specialized quasimode related to Lyapunov exponents, revealing fundamental limits on wave decay in trapped trajectories.

Contribution

It introduces a novel polynomial lower bound on the scattering resolvent linked to Lyapunov exponents, advancing understanding of wave decay in trapping scenarios.

Findings

Polynomial lower bound on scattering resolvent proved

Construction of a quasimode localized on a trapped trajectory

Relation between Lyapunov exponents and decay derivatives

Abstract

We prove a new polynomial lower bound on the scattering resolvent. For that, we construct a quasimode localized on a trajectory $\gamma$ which is trapped in the past, but not in the future. The power in the bound is expressed in terms of the maximal Lyapunov exponent on $\gamma$, and gives the minimal number of derivatives lost in exponential decay of solutions to the wave equation.