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

Hanen Louati (1, 2) , Michel L. Rouleux (1, 2)

1 CPT - E8 Dynamique quantique et analyse spectrale
2 CPT - Centre de Physique Théorique - UMR 7332

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.

Domaines: Physique mathématique [math-ph]

Hanen Louati, Michel L. Rouleux. Semi-classical quantization rules for a periodic orbit of hyperbolic type. Days on Diffraction (DD), 2016, Jun 2016, Saint-Petersbourg, Russia. pp.285-290, ⟨10.1109/DD.2016.7756858⟩. ⟨hal-01591917⟩