Positive commutators, Fermi golden rule and the spectrum of zero temperature Pauli–Fierz Hamiltonians

We perform the spectral analysis of a zero temperature Pauli–Fierz system for small coupling constants. Under the hypothesis of Fermi golden rule, we show that the embedded eigenvalues of the uncoupled system disappear and establish a limiting absorption principle above this level of energy. We rely on a positive commutator approach introduced by Skibsted and pursued by Georgescu–Gérard–Møller. We complete some results obtained so far by Dereziński–Jakšić on one side and by Bach–Fröhlich–Sigal–Soffer on the other side.