From open quantum systems to open quantum maps

In this paper we show that for a class of open quantum systems satisfying a natural dynamical assumption (see §2.2) the study of the resolvent, and hence of scattering, and of resonances, can be reduced, in the semiclassical limit, to the study of open quantum maps, that is of finite dimensional quantizations of canonical relations obtained by truncation of symplectomorphisms derived from the classical Hamiltonian flow (Poincaré return maps). We first explain the result in a simplified setting. For that consider the Schrödinger operator