Semiclassical estimates of the cut-off resolvent for trapping perturbations

TL;DR

This paper provides semiclassical estimates for the resolvent of a black box operator near trapped sets, relating its behavior to regions far from the origin, with implications for resonance states.

Contribution

It introduces new bounds for the resolvent norm near trapped sets without assumptions on the trapped set or resonance multiplicity.

Findings

Resolvent estimates hold with explicit exponential bounds.

Bounds apply to resonance states without additional assumptions.

Results are valid in a specified unphysical sheet region.

Abstract

This paper is devoted to the study of a semiclassical "black box" operator $P$. We estimate the norm of its resolvent truncated near the trapped set by the norm of its resolvent truncated on rings far away from the origin. For $z$ in the unphysical sheet with $- h |ln h| < Im z < 0$, we prove that this estimate holds with a constant $h |Im z|^{-1} e^{C|Im z|/h}$. We also obtain analogous bounds for the resonances states of $P$. These results hold without any assumption on the trapped set neither any assumption on the multiplicity of the resonances.