Domains for Dirac–Coulomb min-max levels

Abstract

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 D_V . 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 D_V , in a range of simple function spaces independent of V . Our results include the critical case lim inf _{x \to 0} |x| V(x)= -1 , with units such that \hbar=mc^2=1 , and they are the first ones in this situation. We also give the corresponding results in two dimensions.