Archive for Rational Mechanics and Analysis

, Volume 192, Issue 3, pp 453–499 | Cite as

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

  • Christian Hainzl
  • Mathieu Lewin
  • Éric Séré


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 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 α is small whereas αZ 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 α 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.


Lewin Quantum Electrodynamic Essential Spectrum Weak Coupling Regime Charge Sector 


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© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA
  2. 2.CNRS and Département de Mathématiques (CNRS UMR 8088)Université de Cergy-PontoiseCergy-Pontoise CedexFrance
  3. 3.CEREMADE (CNRS UMR 7534)Université Paris-DauphineParis Cedex 16France

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