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We consider a hydrogen-like atom in a quantized electromagnetic field whichis modeled by means of a no-pair operator acting in the positive spectralsubspace of the free Dirac operator minimally coupled to the quantized vectorpotential. We prove that the infimum of the spectrum of the no-pair operator isan evenly degenerate eigenvalue. In particular, we show that the bottom of itsspectrum is strictly less than its ionization threshold. These results holdtrue, for arbitrary values of the fine-structure constant and the ultra-violetcut-off and for all Coulomb coupling constants less than the critical one ofthe Brown-Ravenhall model. For Coulomb coupling constants larger than thecritical one, we show that the quadratic form of the no-pair operator isunbounded below. Along the way we discuss the domains and operator cores of thesemi-relativistic Pauli-Fierz and no-pair operators, for Coulomb couplingconstants less than or equal to the critical ones.

Existence of ground states of hydrogen-like atoms in relativistic QED II: The no-pair operator