Resonances for homoclinic trapped sets

TL;DR

This paper investigates semiclassical resonances associated with homoclinic trapped sets, establishing their absence in certain regions, deriving quantization rules, and describing their asymptotic behavior and resonant states.

Contribution

It provides new results on the distribution and asymptotics of resonances for homoclinic and heteroclinic trapped sets, including quantization rules and resonance accumulation patterns.

Findings

No resonances below the real axis under certain conditions

Asymptotic expansion of resonances with finite homoclinic trajectories

Resonances can accumulate on curves or form clouds

Abstract

We study semiclassical resonances generated by homoclinic trapped sets. First, under some general assumptions, we prove that there is no resonance in a region below the real axis. Then, we obtain a quantization rule and the asymptotic expansion of the resonances when there is a finite number of homoclinic trajectories. The same kind of results is proved for homoclinic sets of maximal dimension. Next, we generalize to the case of homoclinic/heteroclinic trajectories and we study the three bump case. In all these settings, the resonances may either accumulate on curves or form clouds. We also describe the corresponding resonant states.