A presentation of a homogenization spectral problem

We present an approach due to Buslaev and Pozharsky of the study of spectral properties of a Schroedinger operator in dimension n. The potential depends of the variable x and the variable x/epsilon. It is periodic in the variable x/epsilon and rapidly decreasing in x. We built asymptotic solutions and we prove the existence of resonances under some assumptions on the homogenized operator.