Fully Coupled Pauli-Fierz Systems at Zero and Positive Temperature

The purpose of these notes is to give a fairly narrow but thorough introduction to the spectral analysis of Hamiltonians and standard Liouvilleans describing finite dimensional small systems linearly coupled to a scalar massless field or reservoir. The Hamiltonians describe the system at zero temperature, and the standard Liouvillean implements unitarily the dynamics of the system at positive temperature. We focus our attention on results valid at arbitrary coupling strength and whose proofs are purely operator theoretic, i.e. for the standard Liouvillean does not make use of the underlying modular structure. This means that important structure results at positive temperature that does not seem to have a purely operator theoretic proof will only be reviewed.