  In this review 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. For both models we discuss the
semi-boundedness of the Hamiltonian, the strict positivity of the ionization
energy, and the exponential localization in position space of spectral
subspaces corresponding to energies below the ionization threshold. Moreover,
we prove the existence of degenerate ground state eigenvalues at the bottom of
the spectrum of the Hamiltonian in both models. All these results hold true,
for arbitrary values of the fine-structure constant and the ultra-violet
cut-off, and for a general class of electrostatic potentials including the
Coulomb potential with nuclear charges less than (sometimes including) the
critical charges without radiation field. Apart from a detailed discussion of
diamagnetic inequalities in QED (which are applied to study the
semi-boundedness) all results stem from earlier articles written by the
authors. While a few proofs are merely sketched, we streamline earlier proofs
or present alternative arguments at many places.
