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(mc2,2αZ−21), 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≤3/4α. Implications for the operators in the partial wave
decomposition are also described.Comment: To appear in Letters in Math. Physic