Existence of ground states of hydrogen-like atoms in relativistic QED I: The semi-relativistic Pauli-Fierz operator

TL;DR

This paper proves the existence of ground states for a semi-relativistic model of hydrogen-like atoms in quantum electrodynamics, showing the spectrum's infimum is an eigenvalue below the ionization threshold under certain conditions.

Contribution

It establishes the existence of ground states for the semi-relativistic Pauli-Fierz operator for hydrogen-like atoms with arbitrary fine-structure constants and cut-offs, given a Coulomb coupling constraint.

Findings

The spectrum's infimum is an eigenvalue.

The ground state energy is below the ionization threshold.

Results hold for all fine-structure constants and cut-offs under the Coulomb coupling limit.

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 ultra-violet cut-off as long as the Coulomb coupling constant (i.e. the product of the fine-structure constant and the nuclear charge) is less than 2/\pi.