Dirac–Coulomb operators with general charge distribution I. Distinguished extension and min-max formulas

Maria J Esteban,Mathieu Lewin,Éric Séré

doi:10.5802/ahl.106

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.

Highlights

Relativistic effects play an important role in the description of quantum electrons in molecules containing heavy nuclei, even for not so large values of the nuclear charge
A proper description of such atoms and molecules is based on the Dirac operator [ELS08, Tha92]
For instance the spectrum of the free Dirac operator is not semi-bounded which prevents from giving an unambiguous definition of a “ground state” and turns out to be related to the existence of the positron [ELS08]

Summary

Introduction

Relativistic effects play an important role in the description of quantum electrons in molecules containing heavy nuclei, even for not so large values of the nuclear charge. In a second step we consider the particular case of a positive measure (or more generally a measure so that the Coulomb potential μ ∗ |x|−1 is bounded from below) and we characterize the domain using a method introduced in [EL07, EL08] and recently generalized in [SST20] This method allows us to provide min-max formulas for the eigenvalues in the gap (−1, 1), following [DES00a, DES00b, DES03, DES06, ELS19, GS99, MM15, Mül[16], SST20]. Μ(R3) ν that is, we ask what is the lowest possible eigenvalue of all possible charge distributions with μ(R3) ν This problem is our main motivation for studying Dirac operators of the type (1.1) with general measures μ. The rest of the paper is devoted to the proofs of our main results

Notation

Distinguished self-adjoint extension

Domain and min-max formulas for positive measures

Description of the domain

Min-max formulas for the eigenvalues

Vμ D0 Vμ