Solutions of the Dirac–Fock Equations and the Energy of the Electron-Positron Field

Technische Universität München, München, Bavaria, Germany
Archive for Rational Mechanics and Analysis (Impact Factor: 2.22). 03/2007; 184(1):1-22. DOI: 10.1007/s00205-006-0016-6


We consider atoms with closed shells, i.e. the electron number N is 2, 8, 10,..., and weak electron-electron interaction. Then there exists a unique solution γ of the Dirac–Fock equations
[Dg,a(g),g]=0[D_{g,\alpha}^{(\gamma)},\gamma]=0 with the additional property that γ is the orthogonal projector onto the first N positive eigenvalues of the Dirac–Fock operator Dg,a(g)D_{g,\alpha}^{(\gamma)}. Moreover, γ minimizes the energy of the relativistic electron-positron field in Hartree–Fock approximation, if the splitting

\mathfrakH:=L2(\mathbbR3)Ä\mathbbC4\mathfrak{H}:=L^2(\mathbb{R}^3)\otimes \mathbb{C}^4 into electron and positron subspace is chosen self-consistently, i.e. the projection onto the electron-subspace is given
by the positive spectral projection ofDg,a(g)D_{g,\alpha}^{(\gamma)}. For fixed electron-nucleus coupling constant g:=α Z we give quantitative estimates on the maximal value of the fine structure constant α for which the existence can be guaranteed.

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    • "Other studied linear and nonlinear models based on the Dirac operator for finitely many particles in an 'inert' vacuum and with a classical electromagnetic field, see e.g. [13] [15] [17] [20] [21] [35] and the references in [16]. Finally, the quantum vacuum and the process of pair creation was investigated in a noninteracting setting (meaning without any light at all), e.g. in [36] [37] [45] [46] [49]. "
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    ABSTRACT: We describe several recent results obtained in collaboration with P. Gravejat, C. Hainzl, E. S\'er\'e and J.P. Solovej, concerning a nonlinear model for the relativistic quantum vacuum in interaction with a classical electromagnetic field.
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    • "Using the contraction principle, they solve an equation analog to (10) with the Laplacian replaced by the Dirac operator. We expect that our methods would allow to improve the results in [12]. Minimization problems for semi-bounded Dirac-Fock type functionals are studied in [10], and in the translation invariant case (no external potential) the minimizer is shown to be unique. "
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    ABSTRACT: In this paper we study the problem of uniqueness of solutions to the Hartree and Hartree-Fock equations of atoms. We show, for example, that the Hartree-Fock ground state of a closed shell atom is unique provided the atomic number $Z$ is sufficiently large compared to the number $N$ of electrons. More specifically, a two-electron atom with atomic number $Z\geq 35$ has a unique Hartree-Fock ground state given by two orbitals with opposite spins and identical spatial wave functions. This statement is wrong for some $Z>1$, which exhibits a phase segregation.
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    • "Lemma IV.12 ([87], "

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