Domains for Dirac–Coulomb min-max levels

We consider a Dirac operator in three space dimensions, with an electrostatic (i.e., real-valued) potential V (x), having a strong Coulomb-type singularity at the origin. This operator is not always essentially self-adjoint but admits a distinguished self-adjoint extension DV . In a first part we obtain new results on the domain of this extension, complementing previous works of Esteban and Loss. Then we prove the validity of min-max formulas for the eigenvalues in the spectral gap of DV , in a range of simple function spaces independent of V . Our results include the critical case lim infx→0 |x|V (x) = −1, with units such that = mc² = 1, and they are the first ones in this situation. We also give the corresponding results in two dimensions.