Hyperbolic Hamiltonian flows and the semi-classical Poincar\'e map

TL;DR

This paper develops a semi-classical framework for analyzing resonances associated with hyperbolic periodic orbits in Schrödinger operators, utilizing the quantization of Poincaré maps in action-angle variables.

Contribution

It introduces a novel analytic approach to semi-classical quantization of Poincaré maps for hyperbolic orbits, advancing the understanding of semi-excited resonances.

Findings

Resonances are characterized via semi-classical quantization of Poincaré maps.

The framework applies to Schrödinger operators with small Planck constant.

Provides a new method for analyzing hyperbolic periodic orbits in quantum systems.

Abstract

We consider semi-excited resonances created by a periodic orbit of hyperbolic type for Schr\"odinger type operators with a small "Planck constant". They are defined within an analytic framework based on the semi-classical quantization of Poincar\'e map in action-angle variables.