Ground state for the relativistic one electron atom

TL;DR

This paper investigates the existence of a ground state for a relativistic one-electron atom modeled by the Dirac-Maxwell system with Coulomb potential, employing a transformation to analyze the problem as an elliptic PDE.

Contribution

It introduces a novel approach using the Foldy--Wouthuysen transformation to study the Dirac-Maxwell system as an elliptic boundary value problem for the first time.

Findings

Existence of a ground state solution established.

Transformation simplifies the analysis of the Dirac-Maxwell system.

Elliptic problem formulation enables new analytical techniques.

Abstract

We study the Dirac-Maxwell system coupled with an external potential of Coulomb type. We use the Foldy--Wouthuysen (unitary) transformation of the Dirac operator and its realization as an elliptic problem in the 4-dim half space $\mathbb{R}^4_{+}$ with Neumann boundary condition. Using this approach we study the existence of a "ground state" solution.