2007/12/18 by Philippe Gravejat, Mathieu Lewin, Éric Séré +1
Mathematics · Physics and Astronomy · #Atomic and Molecular Physics #Cold Atom Physics and Bose-Einstein Condensates #Spectral Theory in Mathematical Physics #math-ph #math.MP
paper · pdf · doi:10.1007/s00220-008-0660-9
37 pages, 1 figure
arxiv created 2007/12/18 · openalex publication_date 2008/11/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the reduced Bogoliubov-Dirac-Fock (BDF) energy which allows to describe relativistic electrons interacting with the Dirac sea, in an external electrostatic potential. The model can be seen as a mean-field approximation of Quantum Electrodynamics (QED) where photons and the so-called exchange term are neglected. A state of the system 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 (the Dirac sea). We study the minimization of the reduced BDF energy under a charge constraint. We prove the existence of minimizers for a large range of values of the charge, and any positive value of the coupling constant α. Our result covers neutral and positively charged molecules, provided that the positive charge is not large enough to create electron-positron pairs. We also prove that the density of any minimizer is an L1 function and compute the effective charge of the system, recovering the usual renormalization of charge: the physical coupling constant is related to α by the formula α\rm phys≃ α(1+2α/(3π)logΛ)-1, where Λ is the ultraviolet cut-off. We eventually prove an estimate on the highest number of electrons which can be bound by a nucleus of charge Z. In the nonrelativistic limit, we obtain that this number is ≤ 2Z, recovering a result of Lieb. This work is based on a series of papers by Hainzl, Lewin, Sere and Solovej on the mean-field approximation of no-photon QED.