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

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2003.04004.pdf (336.0Kb)
Date
2020
Publisher city
Paris
Publisher
Cahier de recherche CEREMADE, Université Paris-Dauphine
Publishing date
03-2020
Link to item file
https://hal.archives-ouvertes.fr/hal-02503460
Dewey
Sciences connexes (physique, astrophysique)
Sujet
Dirac-Coulomb operators
URI
https://basepub.dauphine.fr/handle/123456789/20608
Collections
  • CEREMADE : Publications
Metadata
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Author
Esteban, Maria J.
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Lewin, Mathieu
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Séré, Eric
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Document de travail / Working paper
Item number of pages
33
Abstract (EN)
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 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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