We consider a Dirac operator in three space dimensions, with an electrostatic (i.e., real-valued) potential , having a strong Coulomb-type singularity at the origin. This operator is not always essentially self-adjoint but admits a distinguished self-adjoint extension . 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 , in a range of simple function spaces independent of . Our results include the critical case lim inf, with units such that , and they are the first ones in this situation. We also give the corresponding results in two dimensions.
Cite this article
Maria J. Esteban, Mathieu Lewin, Éric Séré, Domains for Dirac–Coulomb min-max levels. Rev. Mat. Iberoam. 35 (2019), no. 3, pp. 877–924DOI 10.4171/RMI/1074