Volume 4 (2021)
Esteban, Maria J.;  Lewin, Mathieu;  Séré, Éric
Dirac–Coulomb operators with general charge distribution I. Distinguished extension and min-max formulas
Annales Henri Lebesgue, Volume 4 (2021), pp. 1421-1456.

Metadata

DOI10.5802/ahl.106
Keywords Dirac-Coulomb operators, self-adjointness, min-max formulas

Abstract

This paper is the first of a series where we study the spectral properties of Dirac operators with the Coulomb potential generated by any finite signed charge distribution μ. We show here that the operator has a unique distinguished self-adjoint extension under the sole condition that μ has no atom of weight larger than or equal to one. Then we discuss the case of a positive measure and characterize the domain using a quadratic form associated with the upper spinor, following earlier works [EL07, EL08] by Esteban and Loss. This allows us to provide min-max formulas for the eigenvalues in the gap. In the event that some eigenvalues have dived into the negative continuum, the min-max formulas remain valid for the remaining ones. At the end of the paper we also discuss the case of multi-center Dirac–Coulomb operators corresponding to μ being a finite sum of deltas.


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