05-428 Matthias Huber, Heinz Siedentop

Solutions of the Dirac-Fock Equations and the Energy of the Electron-Positron Field (302K, pdf) Dec 20, 05

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Abstract. 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 $\gamma$ of the Dirac-Fock equations $[D_{g,\alpha}^{(\gamma)},\gamma]=0$ with the additional property that $\gamma$ is the orthogonal projector onto the first $N$ positive eigenvalues of the Dirac-Fock operator $D_{g,\alpha}^{(\gamma)}$. Moreover, $\gamma$ minimizes the energy of the relativistic electron-positron field in Hartree-Fock approximation, if the splitting of $\gH:=L^2(\rz^3)\otimes \cz^4$ into electron and positron subspace, is chosen self-consistently, i.e., the projection onto the electron-subspace is given by the positive spectral projection of $D_{g,\alpha}^{(\gamma)}$. For fixed electron-nucleus coupling constant $g:=\alpha Z$ we give quantitative estimates on the maximal value of the fine structure constant $\alpha$ for which the existence can be guaranteed.

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