RUI: Spectral Properties of Some Random Schrodinger Operators

[摘要]舒斌本研究将涉及数学物理方面的问题。提出的几个问题围绕着树或图或连续体上随机薛定谔算子的谱理论。本文还讨论了积分方程研究中的一个应用于物理模型的问题。要研究的一些问题来自固体物理领域。为了形成一个大致的概念，首先想象一个粒子的运动，比如电子在一个具有完美规则结构的晶体中的运动。众所周知，具有一定能量的电子（在带中）会穿过晶体。换句话说，晶体起着导体的作用。考虑一个有杂质的晶体。这样的系统被称为无序的。这种无序的晶体也会充当导体吗？为了开始回答这个问题，需要给无序下一个精确的定义。然而，许多类型的无序晶体不是导体；相反，它们是绝缘体。自从诺贝尔奖得主P. Anderson的开创性工作以来，无序系统一直是数学研究的重点对象。尽管在过去的四十年里许多问题已经得到了解答，但仍有更多的问题有待解决。特别是空间维数大于1的无序系统，人们还没有很好地理解。

Abstract Shubin The research will be concerned with problems from mathematical physics. Several of the proposed problems center around the spectral theory of random Schrodinger operators on trees or graphs or in the continuum. A problem from the study of integral equations which has applications to models from physics is also discussed. Some of the problems to be investigated arise from the field of solid state physics. To form a rough idea, start by imagining the motion of a particle such as an electron in a crystal with a perfectly regular structure. It is well-known that an electron, with a certain energy (in a band), will travel through the crystal. In other words, the crystal acts as a conductor. Consider a crystal with impurities. Such a system is called disordered. Will this disordered crystal also act as a conductor? In order to begin to answer this question, a precise definition of disorder needs to be given. However many types of disordered crystals are not conductors; rather they are insulators. Since the seminal work of Nobel Prize winner P. Anderson, disordered systems have been the object of intense mathematical investigations. Although many questions have been answered in the past forty years, many more problems remain open. In particular, disordered systems with space dimension greater than one, are not well understood.