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MP: Theoretische und Mathematische Grundlagen der Physik

MP 8: Endliche Quantensysteme

MP 8.1: Fachvortrag

Donnerstag, 26. März 1998, 14:00–14:20, H46

Infinite Dimensional Perturbation Theory of Hamilton Operators — •Anja Schlömerkemper and Manfred Requardt —
Institut für Theoretische Physik, Universität Göttingen, Bunsenstraße 9, 37073 Göttingen

The perturbation of a well-behaved Hamiltonian, H₀, by a potential, V, dependent on a coupling parameter λ∈ C 1 is a problem which often comes under consideration in quantum mechanics. Examples from the theory of disordered systems suggest the study of more general perturbations which depend on infinitely many complex parameters or coupling parameters taken from a suitable Banach space. The study of such generalized problems is, however, far from mathematically trivial. It requires the introduction of a fair amount of quite sophisticated mathematics such as complex analysis in Banach spaces. A portion of our work has been devoted to these difficulties and we have succeeded in extending several theorems from the ordinary perturbation theory of operators to this much more general scenario. As a typical example, we treat the model Hamiltonian H(β) = H₀ + ∑_i=1^∞β_i V_i, β = (β_i)_{i∈ I N} ∈ l^∞(C 1) for various classes of potentials V_i.