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

TL;DR

This paper proves the existence of a ground state eigenvalue for a relativistic quantum electrodynamics model of hydrogen-like atoms using a no-pair operator, valid for various physical parameters.

Contribution

It establishes the presence of a ground state eigenvalue for the no-pair operator in relativistic QED models of hydrogen-like atoms across a range of parameters, including the critical Coulomb coupling.

Findings

The spectrum's infimum is an eigenvalue with even degeneracy.

The bottom of the spectrum is below the ionization threshold.

The quadratic form is unbounded below for super-critical Coulomb coupling.

Abstract

We consider a hydrogen-like atom in a quantized electromagnetic field which is modeled by means of a no-pair operator acting in the positive spectral subspace of the free Dirac operator minimally coupled to the quantized vector potential. We prove that the infimum of the spectrum of the no-pair operator is an evenly degenerate eigenvalue. In particular, we show 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 ultra-violet cut-off and for all Coulomb coupling constants less than the critical one of the Brown-Ravenhall model. For Coulomb coupling constants larger than the critical one, we show that the quadratic form of the no-pair operator is unbounded below. Along the way we discuss the domains and operator cores of the semi-relativistic Pauli-Fierz and no-pair operators, for Coulomb coupling constants less than or equal to the critical ones.