  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.
