Challenges in the Theory of Random Schrodinger Operators

随机薛定谔算子理论的挑战

• 批准号: 0503784
• 负责人: Peter Hislop
• 金额: --
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-15 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0503784&HistoricalAwards=false
• 关键词: Challenges; Theory; Random; Schrodinger; Operators

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.

随机薛定谔算子理论的挑战。随机薛定谔算符的谱和输运性质一直是人们深入研究的对象。安德森局域化，即稠密纯点谱的出现几乎是必然的，已被证明在许多模型的带边和谱底。精化的估计给出了关于系统的本征函数的衰减和动力学局部化的精确信息。通过二阶电流-电流关联函数描述了系统的电导率特性，研究这些关联函数可以揭示系统的Mott电导率、态密度和本征值统计等信息。关于这些函数的规律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。随机薛定谔算符提供了一个模型，用于描述电子在被介质中随机分布的杂质破坏的完美晶体结构中的传播。这些模型的研究有望揭示低温下有限电导率和整数量子霍尔效应的机制。新的进展允许一个调查这些模型的传输特性表示通过相关函数。这些相关函数描述了物理上可测量的量，例如态密度和莫特电导率。