Quantization of resonances in the semiclassical limit

This talk will be concerned with the quantum resonances, the poles of the resolvent of the Schroedinger operator. The asymptotic distribution of the resonances close to the real axis in the semiclassical limit is closely related with the trapped trajectories of the underlying classical mechanics. I will begin with the background of the problem and some elementary examples, and end with a very recent work about the resonances created by many homoclinic and heteroclitic trajectories.