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

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 an h-Pseudo-Differential Operator H(y, hDy; h) induced by a periodic orbit of hyperbolic type at energy Eo. We generalize the framework of Gérard, Sjöstrand (1987), in the sense that we allow for both hyperbolic and elliptic eigenvalues of Poincare map, and show that all resonances in W = [Eo − εo, Eo + εo] − i], 0, h^δ], 0 < δ < 1, are given by a generalized Bohr-Sommerfeld quantization rule (BS).