Browsing by Author "Hantsch, Fabian Clemens"

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    The Hartree-Fock equations in quantum mechanics
    (2012) Hantsch, Fabian Clemens; Griesemer, Marcel (Prof. Dr.)
    In the dissertation at hand various aspects of the Hartree-Fock approximation for non-relativistic atoms are considered, in particular the uniqueness of solutions to the Hartree-Fock equations and the existence of ground states in the case of simply charged negative ions. Moreover, we provide new results concerning the related Hartree and Pekar functionals. The first part of this work is devoted to the uniqueness of critical points of the Hartree-Fock functional. It is shown that the minimizer of this functional is unique, if the number of electrons satisfies a certain closed shell condition and the nuclear charge is large enough. Under these assumptions, we also prove the existence and uniqueness of further critical points for the Hartree-Fock functional. Furthermore, we provide a similar result for a restricted Hartree-Fock functional. As an application we obtain a uniqueness result for the minimizer of the Hartree functional. The second part is concerned with several restricted Hartree-Fock functionals, which appear, for example, for closed shell atoms, and we ask whether a minimizer exists for atoms with nuclear charge Z and N electrons. It turns out that the restricted Hartree-Fock functionals for closed shell atoms possess a minimizer, if Z >= N-1. Additionally, we provide sufficient conditions for the existence of minimizers in the case Z = N-1 within the UHF theory. In the third part, we investigate the magnetic Pekar functional. We establish the existence of a minimizer for this functional in the case of a constant magnetic field. Beyond, we consider the situation, where the constant magnetic field is perturbed locally and an external scalar potential is turned on. It turns out that the perturbed Pekar functional possesses a minimizer, if the perturbations are energy reducing. The existence of minimizers allows us to give an easy proof for the binding of two polarons in the model of Pekar and Tomasevich.