Exponential Localization of Hydrogen-like Atoms in Relativistic Quantum Electrodynamics

We consider two different models of a hydrogenic atom in a quantized electromagnetic field that treat the electron relativistically. The first one is a no-pair model in the free picture, the second one is given by the semi-relativistic Pauli-Fierz Hamiltonian. We prove that the no-pair operator is semi-bounded below and that its spectral subspaces corresponding to energies below the ionization threshold are exponentially localized. Both results hold true, for arbitrary values of the fine-structure constant, e ², and the ultra-violet cut-off, Λ, and for all nuclear charges less than the critical charge without radiation field, Z _c  = e ⁻²2/(2/π + π/2). We obtain similar results for the semi-relativistic Pauli-Fierz operator, again for all values of e ² and Λ and for nuclear charges less than e ⁻²2/π.