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Existence of a minimizer for the quasi-relativistic Kohn-Sham model. (English) Zbl 1241.81052

Summary: We study the standard and extended Kohn-Sham models for quasi-relativistic \(N\)-electron Coulomb systems; that is, systems where the kinetic energy of the electrons is given by the quasi-relativistic operator \[ \sqrt{-\alpha^{-2}\Delta_{x_n}+\alpha^{-4}}-\alpha^{-2}. \] For spin-unpolarized systems in the local density approximation, we prove existence of a ground state (or minimizer) provided that the total charge \(Z_{\text{tot}}\) of K nuclei is greater than \(N-1\) and that \(Z_{\text{tot}}\) is smaller than a critical charge \(Z_{\text{c}}=2 \alpha^{-1} \pi^{-1}\).

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
81V55 Molecular physics
81V45 Atomic physics
49S05 Variational principles of physics
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