标题： 散射理论方法在二维随机薛定谔算子中的应用 显示英文标题

标题： Scattering Theory Approach to Random Schroedinger Operators in One Dimension

摘要： 散射理论的方法被引入以通过将体积截断应用于势来分析一维随机薛定谔算子。 关键要素是利夫希茨-克雷因谱移位函数，它通过伯尔曼和克雷因的定理与散射相位相关联。 谱移位密度定义为相互作用区域单位长度的谱移位函数的“热力学极限”。 此密度被证明等于自由哈密顿量和相互作用哈密顿量的状态密度之差。 基于此构造，我们给出了汤普森公式的新的证明。 我们提供了一种从散射矩阵获得李雅普诺夫指数的方法，这表明了如何将这一概念扩展到高维情况。 该方法还允许对具有零李雅普诺夫指数的能量进行表征。 显示英文摘要

摘要： 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.