Scattering Theory Approach to Random Schroedinger Operators in One Dimension

TL;DR

This paper introduces a scattering theory framework for analyzing one-dimensional random Schrödinger operators, connecting spectral shift functions, densities of states, and Lyapunov exponents to deepen understanding of spectral properties.

Contribution

It presents a novel approach using scattering theory and spectral shift functions to analyze spectral properties and Lyapunov exponents in 1D random Schrödinger operators.

Findings

Spectral shift density equals the difference of densities of states.

Provides a new proof of the Thouless formula.

Offers a method to obtain Lyapunov exponents from the scattering matrix.

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.