Research on the Anderson metal-insulator transport transition and otherphenomena in disordered systems

本研究课题为无序系统中的Anderson金属-绝缘体输运跃迁及其他现象。本文将研究一种基于输运而非光谱性质的安德森金属-绝缘体跃迁的新方法。此外，还将研究几个相关主题。将制定在随机媒体中定位的建设性标准；计划了具有随机势的朗道哈密顿量的应用。研究了连续体中安德森型哈密顿量强绝缘子谱的局部泊松统计量。本文将研究贝特晶格上安德森模型的谱。将研究一维或二维低无序定位。自从P. Anderson发表关于随机介质中电子局域化的开创性文章以来，40多年过去了，但我们对金属-绝缘体跃迁的数学理解仍然很不令人满意。在三维或多维度中，人们认为从以局域态为特征的绝缘体状态过渡到以扩展态为特征的非常不同的金属状态。发生这种金属绝缘体跃迁的能量称为迁移率边缘。这幅图的标准数学解释是，随机SchrAdinger的谱中应该有一个从纯点谱（局域态）到绝对连续谱（扩展态）的跃迁。但迄今为止，还没有关于连续谱和金属-绝缘体跃迁存在的数学结果（除了贝特晶格上的安德森模型特例）。提出了一种基于输运而不是光谱性质的安德森金属-绝缘体跃迁的新方法。它的动机是这样一个事实，即局域化的直观物理概念有一个动力学解释：一个最初的局域波包在时间演化中应该保持局域化，而非局域化可以被解释为非平凡输运。本提案的主要目标是证明这种运输过渡的存在。

The subject of this research proposal is the Anderson metal-insulator transport transition and other phenomena in disordered systems. A new approach to the Anderson metal-insulator transition based on transport instead of spectral properties will be investigated. In addition, several related topics will be investigated. Constructive criteria for localization in random media will be developed; an application is planned for the Landau Hamiltonian with a random potential. Local Poisson statistics for the strong insulator spectrum of Anderson-type Hamiltonians in the continuum will be studied. The spectrum of the Anderson model on the Bethe lattice will be studied. Localization at low disorder in one or two dimensions will be investigated.Fortysome years have passed since P. Anderson's seminal article on localization of electrons in random media, but our mathematical understanding of the metal-insulator transition is still very unsatisfactory. In three or more dimensions a transition is believed to occur from an insulator regime, characterized by localized states, to a very different metallic regime characterized by extended states. The energy at which this metal insulator transition occurs is called the mobility edge. The standard mathematical interpretation of this picture is that there should be a transition in the spectrum of the random SchrAdinger from pure point spectrum (localized states) to absolutely continuous spectrum (extended states). But up to now there are no mathematical results on the existence of continuous spectrum and a metal-insulator transition (except for the special case of the Anderson model on the Bethe lattice). A new approach to the Anderson metal-insulator transition is proposed based on transport instead of spectral properties. It is motivated by the fact that the intuitive physical notion of localization has a dynamical interpretation: an initially localized wave packet should remain localized under time evolution, and delocalization may be interpreted as nontrivial transport. The main goal of this proposal is to show the existence of such a transport transition.