Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 306

Scattering Theory Approach to Random Schrödinger Operators in One Dimension

V. Kostrykin and R. Schrader

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 Lifshitz-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 length of the interaction region. This density is shown to be equal to the difference of the densities of states for the free and the interacting Hamiltonians. Based on this construct ion, we give a new proof of the Thouless formula. We provide a prescription how to obtain the Lyapunov exponent from the scattering matrix, which suggest a way how to extend this notion to the higher dimensional case. This prescription also allows a characterization of those energies which have vanishing Lyapunov exponent.

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