Electron. J. Diff. Equ., Vol. 2012 (2012), No. 18, pp. 1-20.

Existence of a minimizer for the quasi-relativistic Kohn-Sham model

Carlos Argaez, Michael Melgaard

Abstract:
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 $Z_{\hbox{tot}}$ of K nuclei is greater than N-1 and that $Z_{\hbox{tot}}$ is smaller than a critical charge $Z_{\hbox{c}}=2 \alpha^{-1} \pi^{-1}$.

Submitted April 12, 2011. Published January 30, 2012.
Math Subject Classifications: 35J60, 47J10, 58Z05, 81V55.
Key Words: Kohn-Sham equations; ground state; variational methods; concentration-compactness; density operators.

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Carlos Argaez
School of Mathematical Sciences
Dublin Institute of Technology
Dublin 8, Ireland
Michael Melgaard
School of Mathematical Sciences
Dublin Institute of Technology
Dublin 8, Ireland
email: mmelgaard@dit.ie

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