Regensburg 1998 – wissenschaftliches Programm
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MP: Theoretische und Mathematische Grundlagen der Physik
MP 1: Hauptvorträge I
MP 1.1: Hauptvortrag
Montag, 23. März 1998, 09:30–10:15, H46
Scattering theory approach to random Schrödinger operators in one dimension — •Robert Schrader1 and V. Kostrykin2 — 1Institut f"ur Theoretische Physik, FU Berlin — 2Lehrstuhl f"ur Lasertechnik, RWTH Aachen
Methods from scattering theory are introduced to analyze random Schrödinger operators in one dimension by applying a volume cutoff to the potential. The key ingredient is the Lifschitz-Krein spectral shift function, which is related to the scattering phase by the theorem of Birman and Krein. The spectral shift density is defined as the thermodynamic limit of the spectral shift function per unit interval. This density is shown to be equal to the difference of the density of states for the free and the interacting Hamiltonian. Based on this construction, we give a new proof of the Thouless formula. We provide a prescription how to obtain the Lyapunov exponent from the scattering matrix, which suggests a way how to extend this notion to the higher dimensional case.