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Properties of Ground States in the Bogoliubov-Dirac-Fock Model

Sok, Jérémy (2013), Properties of Ground States in the Bogoliubov-Dirac-Fock Model. https://basepub.dauphine.fr/handle/123456789/12243

Type
Document de travail / Working paper
External document link
http://hal.archives-ouvertes.fr/hal-00909168
Date
2013
Publisher
Université Paris-Dauphine
Published in
Paris
Pages
43
Metadata
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Author(s)
Sok, Jérémy
Abstract (EN)
We study the Bogoliubov-Dirac-Fock model that allows to consider relativistic electrons interacting with the vacuum in the presence of an external electrostatic field. It can be seen as a Hartree-Fock approximation of QED, where photons are neglected. A state is described by its one-body density matrix: an infinite rank, self-adjoint operator which is a compact perturbation of the negative spectral projector of the free Dirac operator. We are interested in the properties of minimizers of the BDF-energy in the presence of an external field with charge density $\nu\ge 0$ in the regime $\alpha$, $\alpha \log(\Lambda)$ and $\alpha \nu$ (in some norms) small where $\alpha$ is the coupling constant and $\Lambda$ the ultraviolet cut-off. We prove that the density of such minimizer is integrable and compute the effective charge of the system. We also ensure the existence of minimizers under charge constraint $M\in\mathbf{N}^*$ provided that there holds $M-1< \int \nu$ close to the nonrelativistic limit $\alpha\to 0$ with $\alpha\log(\Lambda)$ fixed to a small value. This contrasts with the assumptions of [Arch. Ration. Mech. Anal, 192(3):453-499(2009)] where $\Lambda$ is fixed. As a consequence, the nonrelativistic model we obtain in the limit keeps track of the charge renormalisation: it is different from the Hartree-Fock model obtained.
Subjects / Keywords
ground states; charge renormalisation; Bogoliubov-Dirc-Fock model; Dirac operator

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