Resonances associated to a closed hyperbolic trajectory in dimension 2

TL;DR

This paper analyzes the resonances in a semi-classical setting caused by a closed hyperbolic orbit in two dimensions, providing a complete characterization independent of the semi-classical parameter.

Contribution

It offers a comprehensive description of resonances generated by a closed hyperbolic trajectory, including those from saddle points, in a fixed energy domain.

Findings

All resonances generated by a closed hyperbolic orbit are explicitly determined.

Resonances from a saddle point are characterized within a fixed disc around the critical energy.

The results are independent of the semi-classical parameter.

Abstract

We consider resonances in the semi-classical limit, generated by a single closed hyperbolic orbit, for an operator on ${\bf R}^2$. We determine all such resonancess in a domain independent of the semi-classical parameter As an application we determine all resonances generated by a saddle point in a fixed disc around the critical energy.