  We consider relativistic many-particle operators which - according to Brown
and Ravenhall - describe the electronic states of heavy atoms. Their ground
state energy is investigated in the limit of large nuclear charge and velocity
of light. We show that the leading quasi-classical behavior given by the
Thomas-Fermi theory is raised by a subleading correction, the Scott correction.
Our result is valid for the maximal range of coupling constants, including the
critical one. As a technical tool, a Sobolev-Gagliardo-Nirenberg-type
inequality is established for the critical atomic Brown-Ravenhall operator.
Moreover, we prove sharp upper and lower bound on the eigenvalues of the
hydrogenic Brown-Ravenhall operator up to and including the critical coupling
constant.
