On the virial theorem for the relativistic operator of Brown and Ravenhall, and the absence of embedded eigenvalues

Abstract

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 $\max (m{c}^2,2\alpha Z-\frac{1}{2})$, where $\alpha$ is the fine structure constant, for all values of the nuclear charge $Z$ below the critical value $Z_c$: in particular there are no eigenvalues embedded in the essential spectrum when $Z\le 3/4\alpha$. Implications for the operators in the partial wave decomposition are also described.

To appear in Letters in Math. Physics