Semiclassical Resonances Associated with a Periodic Orbit

We consider resonances for a $h$-pseudo-differential operator $H(x,hD_x;h)$ induced by a periodic orbit of hyperbolic type. We generalize the framework of Gérard and Sjöstrand, in the sense that we allow hyperbolic and elliptic eigenvalues of the Poincaré map, and look for so-called semi-excited resonances with imaginary part of magnitude $-h\log h$, or $h^\delta$, with $0<\delta<1$.