Asymptotic Behavior of Cauchy-Stieltjes Type Integrals of Singular Measures

摘要：Poltoratski的主要研究方向是Cauchy-Stieltjes积分的边界行为。该项目由三部分组成。第一部分主要讨论了复变函数理论中的一些问题。具体来说，PI计划研究柯西积分的奇异和非奇异分量之间的相互作用。该项目的第二部分致力于量子动力学和安德森局域化的应用，这是涉及现代固态物理数学模型的一些最有趣的领域。该项目第三部分的目标是通过对所谓的几乎酉算子的结构的更深入的理解来补充Hilbert空间中收缩算子的完善理论。复变函数理论是现代数学中最经典的部分之一，同时也是现代数学中发展最为迅速的领域之一。它目前的发展阶段的特点是在数学和物理的各个领域有许多有前途的新应用。复变函数理论的基石之一是以法国著名数学家奥古斯丁·柯西命名的积分公式。这个公式的重要性在于，大多数单复变量的解析函数都可以定义为柯西积分或用柯西积分表示。Poltoratski计划研究通过柯西积分在域边界附近定义的函数的各种性质，这些函数的精确行为是微妙的，很难理解。这些结果将被应用于数学中的其他几个中心主题，以及量子动力学。量子动力学的应用将涉及波传播的数学模型。诺贝尔奖得主安德森（P. Anderson）关于无序介质中波的行为的一个猜想，使这一领域近年来得到了广泛的研究。尽管有这些活动，波在无序介质中的传播还没有完全被理解，特别是在大于1的空间维度中。关于这一现象的新信息可能与物理和工程中的许多问题高度相关。

Proposal: DMS-9970151Principal Investigator: Alexei G. PoltoratskiAbstract: Poltoratski's research will focus on a variety of questions related to the boundary behavior of Cauchy-Stieltjes integrals. The project consists of three parts. The first part concentrates on problems in complex function theory. Specifically, the PI plans to study the interaction between singular and nonsingular components of Cauchy integrals. The second part of the project is devoted to applications in quantum dynamics and Anderson localization, some of the most interesting areas involving mathematical models of modern solid state physics. The objective of the third part of the project is to complement the well-developed theory of contractive operators in Hilbert spaces with a deeper understanding of the structure of so-called almost unitary operators.Complex function theory is one of the most classical parts of modern mathematics, yet at the same time is one of its most rapidly expanding and developing fields. The present stage of its development features many promising new applications to various areas of mathematics and physics. Among the cornerstones of complex function theory is the integral formula named after the famous French mathematician Augustin Cauchy. The importance of this formula lies in the fact that most analytic functions of one complex variable can be defined as or represented by Cauchy integrals. Poltoratski plans to study various properties of functions defined via Cauchy integrals near the boundaries of their domains, where the precise behavior of such functions is subtle and difficult to comprehend. The results will then be applied to several other central topics in mathematics, as well as to quantum dynamics. The application to quantum dynamics will deal with mathematical models of wave propagation. A conjecture made by the Nobel laureate P. Anderson about the behavior of waves in disordered media has made this an area of intensive research activity in recent years. Despite this flurry of activity, wave propagation in disordered media is not completely understood, especially in spacial dimensions greater than one. New information about this phenomenon could be highly relevant to a host of problems in physics and engineering.