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

本研究的主题是无序系统中的Anderson金属-绝缘体输运相变等现象。研究了一种基于输运性质而不是光谱性质的安德森金属-绝缘体转变的新方法。此外，还将调查几个相关主题。将制定随机介质中局部化的建设性标准；计划对具有随机势的朗道哈密顿量进行应用。我们将研究连续体中Anderson型哈密顿量强绝缘子谱的局域泊松统计。我们将研究Bethe晶格上Anderson模型的能谱。自P.Anderson关于随机介质中电子局域化的开创性文章发表以来，已经过去了40年，但我们对金属-绝缘体相变的数学理解仍然很不令人满意。在三维或更多维度中，人们认为发生了从以局域态为特征的绝缘体区域到以扩展态为特征的非常不同的金属区域的转变。发生这种金属绝缘体转变的能量称为迁移率边缘。对这幅图的标准数学解释是，随机薛定谔的谱应该有一个从纯点谱(局域态)到绝对连续谱(扩展态)的转变。但到目前为止，除了Bethe晶格上的Anderson模型的特例外，还没有关于连续谱和金属-绝缘体相变的存在的数学结果。提出了一种基于输运性质而不是光谱性质的安德森金属-绝缘体转变的新方法。这是因为局域化的直观物理概念具有动力学解释：初始局域化的波包在时间演化下应该保持局域化，而非局域化可以解释为非平凡的传输。这项提议的主要目的是表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.