A variational approach to the Brown-Ravenhall operator for the relativistic one-electron atoms

TL;DR

This paper introduces a variational method using the Foldy--Wouthuysen transformation to analyze the eigenvalues and eigenfunctions of the Brown-Ravenhall operator in relativistic one-electron atoms, transforming the problem into an elliptic boundary value problem.

Contribution

It provides an alternative characterization of the Brown-Ravenhall operator's spectral problem via a variational approach and elliptic PDE formulation.

Findings

Eigenvalues characterized through elliptic PDE in 4D half-space

Eigenfunctions obtained via variational methods

New boundary condition formulation for the operator

Abstract

We use the Foldy--Wouthuysen (unitary) transformation to give an alternative characterization of the eigenvalues and eigenfunctions for the Brown-Ravenhall operator (the projected Dirac operator) in the case of a one-electron atom. In particular we transform the eigenvalues problem into an elliptic problem in the 4-dim half space $\mathbb{R}^4_{+}$ with Neumann boundary condition.