Semi-classical quantization rules for a periodic orbit of hyperbolic type

TL;DR

This paper develops semi-classical quantization rules for hyperbolic periodic orbits in Hamiltonian systems, extending previous frameworks to include mixed eigenvalues and providing a generalized Bohr-Sommerfeld condition for resonances.

Contribution

It generalizes semi-classical quantization rules to include both hyperbolic and elliptic eigenvalues of the Poincaré map for hyperbolic orbits, broadening the applicability of resonance calculations.

Findings

Resonances are characterized by a generalized Bohr-Sommerfeld rule.

The framework accommodates both hyperbolic and elliptic eigenvalues.

All resonances in the specified energy window are described by this quantization.

Abstract

Determination of periodic orbits for a Hamiltonian system together with their semi-classical quantization has been a long standing problem. We consider here resonances for a $h$-Pseudo-Differential Operator $H(y,hD_y;h)$ induced by a periodic orbit of hyperbolic type at energy $E_0$. We generalize the framework of [G\'eSj], in the sense that we allow for both hyperbolic and elliptic eigenvalues of Poincar\'e map, and show that all resonances in $W=[E_0-\varepsilon_0,E_0+\varepsilon_0]-i]0,h^\delta]$, $0<\delta<1$, are given by a generalized Bohr-Sommerfeld quantization rule.