Vadim Kostrykin, Robert Schrader
Scattering theory approach to random Schroedinger operators in one dimension
(278K, uuencoded Postscript)
ABSTRACT. 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.