On the nature of Fermi Golden Rule for open quantum systems

We consider a general class of models consisting of a small quantum system S interacting with a reservoir R. We compare three applications of 2nd order perturbation theory ( the Fermi Golden Rule) to the study of such models: ( 1) the van Hove ( weak coupling) limit for the dynamics reduced to S; ( 2) the Fermi Golden Rule applied to the Liouvillean - an argument that was used in recent papers on the return to equilibrium; ( 3) the Fermi Golden Rule applied to the so-called C-Liouvillean. These three applications lead to three Level Shift Operators. As our main result, we prove that if the reservoir R is thermal ( if it satisfies the KMS condition), then the Level Shift Operator obtained in ( 1) ( often called the Davies generator) and the Level Shift Operator constructed in ( 2) are connected by a similarity transformation. We also show that the Davies generator coincides with the Level Shift Operator obtained in ( 3) for a general R.