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Journal of Operator Theory

Volume 70, Issue 1, Summer 2013  pp. 211-237.

Ground states of semi-relativistic Pauli-Fierz and no-pair Hamiltonians in QED at critical Coulomb coupling

Authors Martin Koenenberg (1) and Oliver Matte (2)
Author institution: (1) Fakultaet fuer Mathematik und Informatik, FernUniversitaet Hagen, Luetzowstrasse 125, 58084 Hagen, Germany. Current address: Fakultaet fuer Physik, Universitaet Wien, Boltzmanngasse 5, 1090 Vienna, Austria
(2) Fakultaet fuer Mathematik, Technische Universitaet Muenchen, Boltzmannstrasse 3, 85748 Garching, Germany. Current address: Institut for Matematik, Aarhus Universitet, Ny Munkegade 118, 8000 Aarhus C, Denmark

Summary:  We consider the semi-relativistic Pauli--Fierz Hamiltonian and a no-pair model of a hydrogen-like atom interacting with a quantized photon field at the respective critical values of the Coulomb coupling constant. For arbitrary values of the fine-structure constant and the ultra-violet cutoff, we prove the existence of normalizable ground states of the atomic system in both models. This complements earlier results on the existence of ground states in (semi-)relativistic models of quantum electrodynamics at sub-critical coupling by E.~Stockmeyer and the present authors. Technically, the main new achievement is an improved estimate on the spatial exponential localization of low-lying spectral subspaces yielding uniform bounds at large Coulomb coupling constants. In the semi-relativistic Pauli--Fierz model our exponential decay rate given in terms of the binding energy reduces to the one known from the electronic model when the radiation field is turned off. In particular, an increase of the binding energy due to the radiation field is shown to improve the localization of ground states.

Keywords:  existence of ground states, exponential localization, semi-relativistic Pauli-Fierz, no-pair Hamiltonian, quantum electrodynamics

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