On the Virial Theorem for the Relativistic Operator of Brown and Ravenhall, and the Absence of Embedded Eigenvalues

A virial theorem is established for the operator proposed by Brown and Ravenhall as a model for relativistic one-electron atoms. As a consequence, it is proved that the operator has no eigenvalues greater than max(2αZ - \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \frac{1}{2} $$ \end{document})mc2, where α is the fine structure constant, for all values of the nuclear charge Z below the critical value Zc: in particular, there are no eigenvalues embedded in the essential spectrum when Z ≤ 3/4 α. Implications for the operators in the partial wave decomposition are also described.