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

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 $H(y,h{D}_y;h)$ induced by a periodic orbit of hyperbolic type at energy $E_0$. We generalize the framework of [GéSj], in the sense that we allow for both hyperbolic and elliptic eigenvalues of Poincaré map, and show that all resonances in $W=[E_0-{\epsilon }_0,E_0+{\epsilon }_0]-i]0,h^{\delta }]$, $0<\delta <1$, are given by a generalized Bohr-Sommerfeld quantization rule.

arXiv admin note: substantial text overlap with arXiv:1606.06180