Semiclassical Resonances via Meromorphy of the Resolvent and the S-Matrix

The purpose of this paper is to describe the basic problems of resonances via meromorphic continuation of the resolvent and the scattering matrix. An example from mathematical physics is given by investigating the poles of the resolvent of semiclassical Schr ¨odinger operators and Born-Oppenheimer Hamiltonians. Mathematical techniques, dilation-analyticity and Feshbach reduction are used here for the characterization of resonances of these Hamiltonians.