  We consider a one-dimensional effective quantum electrodynamics (QED) model
of the relativistic hydrogen-like atom using delta-potential interactions. We
discuss the general exact theory and the Hartree-Fock approximation. The
present one-dimensional effective QED model shares the essential physical
feature of the three-dimensional theory: the nuclear charge polarizes the
vacuum state (creation of electron-positron pairs) which results in a QED
Lamb-type shift of the bound-state energy. Yet, this 1D effective QED model
eliminates some of the most serious technical difficulties of the
three-dimensional theory coming from renormalization. We show how to calculate
the vacuum-polarization density at zeroth order in the two-particle interaction
and the QED Lamb-type shift of the bound-state energy at first order in the
two-particle interaction. The present work may be considered as a step toward
the development of a quantum-chemistry effective QED theory of atoms and
molecules.
