EXISTENCE OF GROUND STATES OF HYDROGEN-LIKE ATOMS IN RELATIVISTIC QED I: THE SEMI-RELATIVISTIC PAULI–FIERZ OPERATOR

Abstract

We consider a hydrogen-like atom in a quantized electromagnetic field which is modeled by means of the semi-relativistic Pauli–Fierz operator and prove that the infimum of the spectrum of the latter operator is an eigenvalue. In particular, we verify that the bottom of its spectrum is strictly less than its ionization threshold. These results hold true, for arbitrary values of the fine-structure constant and the ultraviolet cut-off as long as the Coulomb coupling constant is less than 2/π. For Coulomb coupling constants larger than 2/π, we show that the quadratic form of the Hamiltonian is unbounded below.