Boundary Behavior of Analytic Functions

本论文的主要研究内容是解析函数的边界性质及其在泛函分析和物理学中的应用。在解析函数论中，PI计划研究复域边界附近柯西积分的奇异分量和非奇异分量之间的相互作用.然后将所得结果应用于研究扰动问题中线性算子的预解函数。这一领域与固体物理学的数学模型有关，这提供了大部分的动力。本项目的另一部分是研究Beurling-Malliavin理论、偏微分方程的反谱问题和Toepliz算子核的性质之间的联系。解析函数论是现代数学中经典但发展迅速的部分之一。它目前的发展阶段的特点是在数学和物理学的各个部分有许多有前途的新应用。解析函数论的基石之一是以法国著名数学家柯西命名的积分公式，该公式的重要性在于复数域上的大多数解析函数都可以通过柯西积分来定义。PI计划研究柯西积分在其区域边界附近的各种性质。这些结果将被应用于数学的几个领域以及数学物理。在数学物理中的应用将涉及波传播的数学模型，一般的问题是如何从波算符的谱和位场的部分信息中恢复有关无序现象的全部信息。

The research will focus on several problems concerningthe boundary behavior of analytic functions and theirapplications in functional analysis and mathematicalphysics. In analytic function theory the PI plans tostudy the interaction between singular and non-singularcomponents of Cauchy integrals near the boundary of acomplex domain. The results will then be applied tostudy the resolvent functions of linear operators inperturbation problems. This area has connections withmathematical models of solid state physics, whichprovides most of the motivation. Another part of theproject is the study of the connections between theBeurling-Malliavin theory, inverse spectral problemsfor partial differential equations and properties ofkernels of Toepliz operators.Analytic Function Theory is one of the classical yetrapidly developing parts of modern mathematics. Thepresent stage of its development features manypromising new applications in various parts ofmathematics and physics. One of the cornerstones ofAnalytic Function Theory is the integral formulanamed after the famous French mathematician Cauchy.The importance of this formula is due to the factthat most analytic functions in complex domains canbe defined through Cauchy integrals. The PI plansto study various properties of Cauchy integrals nearthe boundaries of their domains. The results willthen be applied in several areas of mathematics aswell as in mathematical physics. The applicationsin mathematical physics will concern, among otherthings, mathematical models of wave propagation.The general question that will be considered is howto recover full information about a disorderedenvironment from the spectrum of the wave operator andpartial information on the potential field.