RELATIVISTIC EXTENSION OF THE BETHE SUM-RULE

We have derived an expression in terms of eigenvalues and eigenfunctio ns of the no-pair Hamiltonian for the sum over the generalized oscilla tor strength distribution for the relativistic, positive-energy soluti ons for a many-electron system. Formally, it looks very much like the corresponding nonrelativistic sum. However, only in the nonrelativisti c limit is the sum equal to the number of electrons (the Bethe sum rul e). We have determined the leading (1/c(2)) relativistic correction to the nonrelativistic Bethe sum rule and we argue that this correction is more in accordance with relativistic quantum mechanics than is a pr evious suggestion. The same statement holds for relativistic correctio ns to the Thomas-Reiche-Kuhn sum rule, previously derived only for one -electron systems. We discuss the problem that the explicit summation of the generalized oscillator strengths and calculation of the sum as a groundstate average of a comutator, while formally equivalent, may n ot give identical results in approximate calculations.