Complex and Harmonic Analysis in Spectral Theory; Cyclic and Subcyclic vectors of Rank One Perturbations and Anderson-type Hamiltonians

谱理论中的复数和调和分析；

• 批准号: 1101477
• 负责人: Constanze Liaw
• 金额: $ 10.36万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-15 至 2012-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101477&HistoricalAwards=false
• 关键词: Complex; Harmonic; Analysis; Spectral; Theory

This proposal consists of several parts. Part one concerns Aleksandrov-Clark Theory, which relates rank one perturbations to functional models, the Hilbert transform, holomorphic composition operators, rigid functions and the Nehari interpolation problem. A formula (a generalization of the normalized Cauchy transform) for the adjoint of the Clark operator has recently been proven by the principal investigator in collaboration with Salem Prize winner S. Treil. It should be possible to gain some control over the embedded singular spectrum for rank one perturbations, a long standing problem in the field. Part two pertains to the interpretation of a wide class of singular integral operators, including those of Calderon-Zygmund type, in the two-weight situation with very general measures (in particular, non-doubling). Singular integral operators play an essential role in modern Analysis. The last two parts are devoted to aspects of cyclicity. Cyclicity is related to the question for which measures the corresponding Hardy space is dense in that of square integrable functions with respect to the measure, to the backward shift operator, and a classical result of Douglas, Shapiro and Shields connects cyclicity with pseudocontinuation; and thus opening a new area of complex function theory. In part three, the goal is to prove that (assuming cyclicity of the operator) any non-zero vector yields cyclic vectors for rank one perturbations for almost all parameters. This may be useful in practice. It should be mentioned, that for a cyclic operator, it may not be easy to find a cyclic vector. Two interesting new notions are introduced for the setting of an operator on a separable Hilbert space: Subcyclic vectors (a refinement of the definition of cyclic vectors) and a certain graph. Apart from studying their properties, a deep relationship between them in the context of Anderson-type Hamiltonians has preliminarily been proven by the principal investigator in collaboration with E. Abakumov and A. Poltoratski. The last part of this project is connected to the famous problem of Anderson localization, which was suggested by Nobel laureate P. W. Anderson in 1958. The PI will study the cyclicity of vectors for Anderson-type Hamiltonians (a generalization of most Anderson models, e.g. random Schroedinger operators) via analytical as well as numerical methods.The underlying goal is to develop the mathematical tools necessary to understand the dynamics of physical systems. Such systems are often described by second-order differential equations, like the Schroedinger equation from quantum mechanics (the governing mechanics at the molecular level) and the string equation which is the cutting edge attempt to unite quantum mechanics with general relativity (the theory describing gravity). One of the objects of study - 'singular integral operators' - have become a useful tool in perturbation theory. The latter is concerned roughly with the following question: Given certain information about a physical system, can one predict what happens in the case where one parameter, for example in the initial condition, is changed/perturbed? Cyclicity, for physicists, means that the spectrum (e.g. of light) is simple or non-degenerate. In many problems it is important to know whether this is the case or not. For example, the above-mentioned Anderson localization addresses the question whether or not an impure crystal allows the diffusion of waves or, roughly speaking, whether all electrons stay within a bounded region in space. The results obtained in the scope of this project will be published in scientific journals and reported at research conferences. The proposed subjects provide a wealth of accessible research questions for undergraduate and graduate students. The principal investigator will write expository articles, give seminars at the student level, and mentor young researchers interested in the many open problems she will make available.

这个建议由几个部分组成。第一部分介绍了Aleksandrov-Clark理论，该理论将秩一扰动与泛函模型、Hilbert变换、全纯复合算子、刚性函数和Nehari插值问题联系起来。克拉克算子伴随的一个公式（规范化柯西变换的推广）最近由首席研究员与塞勒姆奖得主S. Treil合作证明。它应该有可能获得一些控制嵌入奇异谱的一级扰动，一个长期存在的问题，在该领域。第二部分解释了一类广义的奇异积分算子，包括Calderon-Zygmund型的奇异积分算子，在具有非常一般测度（特别是非加倍）的双权情况下。奇异积分算子在现代分析中起着重要的作用。后两部分是关于周期性的。环性与对应的Hardy空间在平方可积函数的空间中相对于该测度稠密的问题有关，与倒移算子有关，道格拉斯、夏皮罗和希尔兹的经典结果将环性与伪延拓联系起来；从而开辟了复变函数理论的一个新领域。在第三部分中，目标是证明（假设算子的循环性）对于几乎所有参数的秩一扰动，任何非零向量都产生循环向量。这在实践中可能有用。需要指出的是，对于一个循环算符，找到一个循环向量可能并不容易。引入了两个有趣的新概念：次循环向量（循环向量定义的一种改进）和某一图。除了研究它们的性质外，首席研究员与E. Abakumov和a . Poltoratski合作，初步证明了它们之间在安德森型哈密顿量背景下的深刻关系。这个项目的最后一部分与著名的安德森本地化问题有关，这个问题是由诺贝尔奖获得者p.w.安德森在1958年提出的。PI将通过解析和数值方法研究安德森型哈密顿量（大多数安德森模型的推广，例如随机薛定谔算子）向量的循环性。潜在的目标是开发必要的数学工具来理解物理系统的动力学。这样的系统通常用二阶微分方程来描述，比如量子力学中的薛定谔方程（分子水平上的控制力学）和弦方程，弦方程是将量子力学与广义相对论（描述引力的理论）结合起来的前沿尝试。研究对象之一的“奇异积分算子”已经成为微扰理论中的一个有用的工具。后者大致涉及以下问题：给定关于物理系统的某些信息，人们能否预测当一个参数（例如初始条件中的参数）被改变/扰动时会发生什么？对物理学家来说，周期性意味着光谱（如光）是简单的或非简并的。在许多问题中，了解情况是否如此是很重要的。例如，上面提到的安德森局域化解决了不纯晶体是否允许波扩散的问题，或者，粗略地说，是否所有的电子都停留在空间的一个有界区域内。在本项目范围内获得的结果将在科学期刊上发表，并在研究会议上报告。拟议的科目为本科生和研究生提供了丰富的可访问的研究问题。首席研究员将撰写说明性文章，在学生层面上举办研讨会，并指导对她将提供的许多开放问题感兴趣的年轻研究人员。