We consider a system of finitely many nonrelativistic, quantum mechanical electrons bound to static nuclei. The electrons are minimally coupled to the quantized electromagnetic field; but we impose an ultraviolet cutoff on the electromagnetic vector potential appearing in covariant derivatives, and the interactions between the radiation field and electrons localized very l:ar from the nuclei are turned off. For a class of Hamiltonians we prove exponential localization of bound states, establish the existence of a ground state, and derive sufficient conditions for its uniqueness. Furthermore, we show that excited bound states of the unperturbed system become unstable and turn into resonances when the electrons are coupled to the radiation field. To this end we develop a novel renormalization transformation which acts directly on the space of Hamiltonians. o m e n a t h a t s t o o d at t h e o r i g i n o f q u a n t u m t h e o r y : t h a t o f e m i s s i o n *

The familiar unrestricted Hartree-Fock variational principle is generalized to include quasi-free states. As we show, these are in one-to-one correspondence with the one-particle density matrices and these, in turn provide a convenient formulation of a generalized Hartree-Fock variational principle, which includes the BCS theory as a special case. While this generalization is not new, it is not well known and we begin by elucidating it. The Hubbard model, with its particle-hole symmetry, is well suited to exploring this theory because BCS states for the attractive model turn into usual HF states for the repulsive model. We rigorously determine the true, unrestricted minimizers for zero and for nonzero temperature in several cases, notably the half-filled band. For the cases treated here, we can exactly determine all broken and unbroken spatial and gauge symmetries of the Hamiltonian.

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We consider systems of static nuclei and electrons {atoms and molecules{ coupled to the quantized radiation eld. The interactions between electrons and the soft modes of the quantized electromagnetic eld are described by minimal coupling,p !p eÃ(x), whereÃ(x) i s Heisenberg Fellow of the DFG, supported by SFB 288 of the DFG, the TMR-Network on \PDE and QM". y Supported by NSERC Grant NA 7901 0 EXT-2000-055 01/11/1998 whose lifetimes we estimate. Furthermore the energy spectrum is absolutely continuous, except, perhaps, in a small interval above the ground state energy and around the threshold energies of the atom or molecule.

In this paper we present a self-contained and detailed exposition of the new renormalization group technique proposed in [1,2]. Its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator, the partition function, or correlation functions.We apply this renormalization transformation to a Hamiltonian describing the physics of an atom interacting with the quantized electromagnetic field, and we prove that excited atomic states turn into resonances when the coupling between electrons and field is nonvanishing. Academic Press