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

这项提案由几个部分组成。第一部分是Aleksandrov-Clark理论，它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。最近，首席研究员与塞勒姆奖获得者S.Treil合作证明了Clark算子的伴随公式(归一化柯西变换的推广)。应该有可能对一阶扰动的嵌入奇异谱进行一些控制，这是该领域的一个长期存在的问题。第二部分是关于一大类奇异积分算子的解释，包括Calderon-Zygmund类型的奇异积分算子在具有非常一般的度量(特别是非加倍)的双权情形下的解释。奇异积分算子在现代分析中起着至关重要的作用。最后两部分致力于周期性的各个方面。循环性涉及到相应的Hardy空间对于哪些测度是稠密的，即平方可积函数关于该测度的稠密性，与后移算子有关，Douglas，Shapiro和Shields的一个经典结果将循环性与伪连续性联系起来，从而开辟了复函数理论的一个新领域。在第三部分中，我们的目的是证明(假设算子的循环性)，对于几乎所有参数的一阶扰动，任何非零向量都产生循环向量。这在实践中可能是有用的。值得一提的是，对于循环算子来说，找到循环向量可能并不容易。对于可分Hilbert空间上算子的设置，引入了两个有趣的新概念：亚循环向量(循环向量定义的改进)和某个图。除了研究它们的性质外，主要研究者还与E.Abakumov和A.Poltoratski合作，初步证明了它们之间在Anderson型哈密顿量背景下的深层次联系。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将通过分析和数值方法研究Anderson型哈密顿量(大多数Anderson模型的推广，例如随机薛定谔算子)的向量的循环性。基本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，比如量子力学的薛定谔方程(分子水平上的支配力学)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)相结合的前沿尝试。其中一个研究对象--“奇异积分算子”--已经成为微扰理论中的一个有用工具。后者大致涉及以下问题：给定关于物理系统的某些信息，当一个参数，例如在初始条件中，发生变化/扰动时，人们能预测会发生什么吗？对物理学家来说，周期性意味着光谱(例如光)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，上述安德森定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，是否所有电子都留在空间的有界区域内。在该项目范围内取得的成果将在科学期刊上发表，并在研究会议上报告。拟议的主题为本科生和研究生提供了丰富的可访问的研究问题。首席研究人员将撰写说明性文章，在学生层面上进行研讨会，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。

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.