Existence and construction of resonances for atoms coupled to the quantized radiation field

The processes of emission and absorption of photons by atoms can be rigorously understood in the low-energy limit if we neglect the creation and annihilation of electrons. They are related to eigenvalues of the atomic Hamiltonian that are embedded in the continuous spectrum of the free Hamiltonian of the electromagnetic field. The mathematical analysis of the perturbation of these eigenvalues due to the electromagnetic interaction relies on a complex deformation technique relating the original Hamiltonian to a non-selfadjoint operator. We develop a new technique to analyze the spectrum of operators used in non-relativistic quantum electrodynamics. Our method can be applied to prove most of the results that previously required an involved renormalization group construction. We use analytic perturbation theory of operators in Hilbert spaces instead. More precisely, we extend the multi-scale analysis introduced by one of us in 2003 [15], which was used so far only for the study of selfadjoint operators, to non-selfadjoint operators. Compared to the selfadjoint case (see, for example, [3]) the analysis of these non-selfadjoint operators is more difficult, because we cannot make use of the functional calculus (spectral theorem) and the min-max principle in some crucial estimates.