Scattering theory approach to random Schrodinger operators in one dimension

Methods from scattering theory are introduced to analyze random Schrodinger operators in one dimension by applying a Volume cutoff to the potential. T he key ingredient is the Lifshitz-Krein spectral shift function, which is r elated to the scattering phase by the theorem of Birman and Krein. The spec tral shift density is defined as the "thermodynamic limit" of the spectral shift function per unit length of the interaction region. This density is s hown 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 e xtend this notion to the higher dimensional case. This prescription also al lows a characterization of those energies which have vanishing Lyapunov exp onent.