Semi-classical resonances associated with a periodic orbit of hyperbolic type

Abstract

We consider in this Note resonances for a $h$-Pseudo-Differential Operator $H(x,h{D}_x;h)$ on $L^2(M)$ induced by a periodic orbit of hyperbolic type, as arises for Schrödinger operator with AC Stark effect when $M={\mathbf{R}}^n$, or the geodesic flow on an axially symmetric manifold $M$, extending Poincaré example of Lagrangian systems with 2 degrees of freedom. We generalize the framework of [GéSj], in the sense that we allow for hyperbolic and elliptic eigenvalues of Poincaré map, and look for so-called semi-excited resonances with imaginary part of magnitude $-h\log h$, or $h^s$, with $0<s<1$.