Challenges in the Theory of Random Schrodinger Operators

随机薛定谔算子理论的挑战肯塔基州大学彼得·D·希斯洛摘要随机薛定谔算子的谱和输运性质一直是人们密切研究的对象。Anderson局部化，几乎肯定会出现密集的纯点谱，已经在许多模型的带边和谱底得到了证明。精细估计提供了关于特征函数的衰减和系统的动态局部化的精确信息。通过二阶电流-电流关联函数描述了系统的电导特性，对这些关联函数的研究揭示了关于Mott电导率、态密度和本征值统计的信息。关于这些函数的正则性和界限，还有许多悬而未决的问题。例如，电流-电流相关函数的下限意味着离域。研究这些系统的另一个新工具是使用随机矩阵理论。随机薛定谔算符为电子在被随机分布在介质中的杂质破坏的完美晶体结构中的传播提供了一个模型。希望通过对这些模型的研究，揭示低温下有限电导率和整数量子霍尔效应的机制。新的进展使人们能够研究通过关联函数表示的这些模型的输运性质。这些关联函数描述了物理上可测量的量，如态密度和莫特电导率。

Challenges in the theory of random Schrodinger operatorsPeter D. HislopUniversity of KentuckyAbstractThe spectral and transport properties of random Schrodinger operators have been the object of intense study. Anderson localization, the occurrence of dense pure point spectrum almost surely, has been proved for many models at band-edges and at the bottom of the spectrum. Refined estimates give precise information about the decay of the eigenfucntions and the dynamical localization of the system. Conductivity properties of the system are described through the second-order current-current correlation function.Study of these correlation functions reveal information about Mott conductivity, the density of states, and eigenvalue statistics. There are many open questions about the regularity and bounds on these functions. A lower bound on the current-current correlation function implies delocalization, for example. Another new tool for the study of these systems is the use of random matrix theory. This promises to give insight into the density of states in the delocalized regime.Random Schrodinger operators provide a model for the propagation of electrons in perfect crystalline structures that are corrupted by impurities randomly distributed in the medium. It is hoped that the study of these models reveals the mechanisms for finite conductivity at low temperatures and the integer quantum Hall effect. New advances allow one to investigate the transport properties of these models as expressed through correlation functions. These correlations functions describe physically measurable quantities such as the density of states and the Mott conductivity.