1999/02/01 by Vadim Kostrykin, Robert Schrader · 5 citations
Mathematics · Physics and Astronomy · #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #Theoretical and Computational Physics #math-ph #math.MP #math.SP #msc:34F05 #msc:60H25 #msc:82B44
paper · pdf · doi:10.1142/s0129055x99000088
published as Reviews in Marthematical Physics 11 (1999) 187 -- 242 · 1 figure
openalex publication_date 1999/02/01 · arxiv created 2000/11/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.