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Article Dans Une Revue Archive for Rational Mechanics and Analysis Année : 2009

Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics

Résumé

The Bogoliubov-Dirac-Fock (BDF) model is the mean-field approximation of no-photon Quantum Electrodynamics. The present paper is devoted to the study of the minimization of the BDF energy functional \emph{under a charge constraint}. An associated minimizer, if it exists, will usually represent the ground state of a system of $N$ electrons interacting with the Dirac sea, in an external electrostatic field generated by one or several fixed nuclei. We prove that such a minimizer exists when a binding (HVZ-type) condition holds. We also derive, study and interpret the equation satisfied by such a minimizer. Finally, we provide two regimes in which the binding condition is fulfilled, obtaining the existence of a minimizer in these cases. The first is the weak coupling regime for which the coupling constant $\alpha$ is small whereas $\alpha Z$ and the particle number $N$ are fixed. The second is the non-relativistic regime in which the speed of light tends to infinity (or equivalently $\alpha$ tends to zero) and $Z$, $N$ are fixed. We also prove that the electronic solution converges in the non-relativistic limit towards a Hartree-Fock ground state.

Dates et versions

hal-00132407 , version 1 (21-02-2007)

Identifiants

Citer

Christian Hainzl, Mathieu Lewin, Eric Séré. Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics. Archive for Rational Mechanics and Analysis, 2009, 192 (3), pp.453-499. ⟨10.1007/s00205-008-0144-2⟩. ⟨hal-00132407⟩
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