Scattering Theory Approach to Random Schroedinger Operators in One Dimension

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dc.creatorKostrykin, Vadim
dc.creatorSchrader, Robert
dc.date2000-11-18
dc.date.accessioned2026-07-07T04:28:08Z
dc.date.available2026-07-07T04:28:08Z
dc.descriptionMethods 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.
dc.description1 figure
dc.identifierhttps://arxiv.org/abs/math-ph/0011032
dc.identifierhttp://arxiv.org/abs/math-ph/0011032
dc.identifierReviews in Marthematical Physics 11 (1999) 187 -- 242
dc.identifierdoi:10.1142/S0129055X99000088
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/56669
dc.subjectMathematical Physics
dc.subjectSpectral Theory
dc.subject82B44; 34F05; 60H25
dc.titleScattering Theory Approach to Random Schroedinger Operators in One Dimension
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