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

这项建议包括几个部分。第一部分涉及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。克拉克算子伴随的一个公式（规范化柯西变换的推广）最近已被首席研究员与塞勒姆奖赢家S。特瑞尔秩一扰动的嵌入奇异谱的控制是可能的，这是该领域长期存在的问题。第二部分涉及到解释的一类广泛的奇异积分算子，包括Calderon-Zygmund型，在两个重量的情况下非常一般的措施（特别是，非加倍）。奇异积分算子在现代分析中起着重要的作用。最后两部分专门讨论周期性的各个方面。循环性涉及到的问题是，相应的哈代空间的措施是密集的平方可积函数的措施，以向后移动运营商，和一个经典的结果道格拉斯，夏皮罗和希尔兹连接循环性与伪延续，从而开辟了一个新的领域复变函数理论。在第三部分中，我们的目标是证明（假设运营商的循环）任何非零向量产生循环向量秩1扰动几乎所有的参数。这在实践中可能是有用的。应该提到的是，对于循环算子，可能不容易找到循环向量。在可分希尔伯特空间上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）和某个图。除了研究它们的性质外，主要研究者与E. Abakumov和A.波尔托拉茨基本课题的最后一部分是关于诺贝尔奖获得者P.W.提出的著名的安德森局部化问题。1958年的安德森。PI将通过分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）的向量循环性。基本目标是开发理解物理系统动力学所需的数学工具。这样的系统通常由二阶微分方程描述，如量子力学中的薛定谔方程（分子水平上的支配力学）和弦方程，后者是将量子力学与广义相对论（描述引力的理论）统一起来的前沿尝试。研究的对象之一- '奇异积分算子' -已成为一个有用的工具，在微扰理论。后者大致涉及以下问题：给定关于物理系统的某些信息，人们能否预测在一个参数（例如初始条件）被改变/扰动的情况下会发生什么？对于物理学家来说，周期性意味着光谱（例如光的光谱）是简单的或非简并的。在许多问题中，知道是否是这样是很重要的。例如，上面提到的安德森局域化解决了一个问题，即不纯的晶体是否允许波的扩散，或者粗略地说，是否所有的电子都停留在空间中的有界区域内。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。首席研究员将撰写简要的文章，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。

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.