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

Abstract

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 (HVZtype) 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.

The Hartree Equation for Infinitely Many Particles I. Well-Posedness Theory

“…Our result requires some stringent conditions (26), (27) and (28) on f and w, which are not optimal and which we have not tried to optimize. They are satisfied if for instance f is bounded and decays fast enough, and if w is in the Schwartz class.…”

“…Let d 1, 1 p q < ∞ and s r > d(q − 1)/q or s r 0 if p = q = 1. Assume that w and f satisfy the same assumptions (26), (27) and (28) as in Theorem 3. Then, for any T 1 , (20), we have the following estimate…”

“…Finally, the unique solution Q(t) depends continuously on Q 0 ∈ S p,s , and on w and f : if Q (n) 0 → Q 0 strongly in S p,s , w n → w and f n → f for the norms corresponding to (26), (27) and (28), then the unique solution Q (n) (t) has maximal times of existence T ± n such that lim inf n→∞ T ± n T ± and Q (n) …”

Orbitally Stable States in Generalized Hartree–fock Theory

“…The proof of Theorem 2.1 follows some classical ideas which have been introduced in various papers 24,25,4,18,19 .…”

“…Modifying {γ n } n∈N if necessary, we can assume that {γ n } n∈N is of finite rank and that (−∆) γ n ∈ S 1 for all n . Now we follow a truncation method 18,19 . Let us choose two C ∞ functions χ and ξ with values in [0, 1] such that χ 2 + ξ 2 = 1 , χ has its support in B(0, 2) and χ ≡ 1 on B(0, 1) .…”