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

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.