Scattering Theory Approach to Random Schroedinger Operators in One Dimension

Details Scattering Theory Approach to Random Schroedinger Operators in One Dimension Scattering Theory Approach to Random Schroedinger Operators in One Dimension

2000-11-18

arxiv.org/abs/math-ph/0011032v1

Reviews in Marthematical Physics 11 (1999) 187 -- 242

Physics

Mathematical Physics

1 figure

Scientific paper

10.1142/S0129055X99000088

Methods from scattering theory are introduced to analyze random Schroedinger 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 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 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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