Finite Rank Perturbations and Model Theory

有限阶扰动和模型理论

• 批准号: 1802682
• 负责人: Constanze Liaw
• 金额: $ 12.9万
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802682&HistoricalAwards=false
• 关键词: Finite; Rank; Perturbations; Model; Theory

Many physical systems are modeled by differential operators. One way of describing a system's long-term behavior is through spectral theory, which includes the finding of frequencies naturally exhibited by the system. Imagine a vibrating string or beam of fixed length. Clearly, its frequency (think "sound") depends on how its ends (aka boundaries) are clamped down or otherwise restricted. In general, the spectrum of a differential operator and with it the properties of a physical system can change drastically when the conditions imposed on the boundary are changed. In many cases, we know the complete spectrum for one set of boundary conditions. From there, we can gain knowledge about the system under other conditions via the theory of so-called finite rank perturbations. The primary goal of the project is to systematically study spectral theory of finite rank unitary perturbations by tightening the relationship to corresponding functional models. The rank one setting is reasonably well-understood, while a general treatment of the finite rank problem presented several digressions. Initially, non-cyclic unitary unperturbed operators posed an issue. This problem is now resolved. The general problem involves matrix-valued Herglotz and Cauchy-type transforms. The study of these transforms as part of this project will provide insight into finite rank perturbation theory. A regularization of the so-called exterior Cauchy transform (as was done in the rank one setting) is planned. Further investigations will be made into the difficult case of infinite rank perturbations with defect operators in the trace class. The analogous self-adjoint setting, as well as concrete applications to differential operators, will also be investigated.

许多物理系统都是用微分算子来模拟的。描述系统长期行为的一种方法是通过频谱理论，其中包括发现系统自然表现出的频率。想象一根固定长度的振动弦或梁。显然，它的频率（想想“声音”）取决于它的两端（又名边界）是如何被钳制或以其他方式限制的。一般来说，当施加在边界上的条件改变时，微分算子的谱以及物理系统的性质会发生急剧变化。在许多情况下，我们知道一组边界条件的完整谱。从那里，我们可以通过所谓的有限秩扰动理论获得关于系统在其他条件下的知识。该项目的主要目标是通过加强与相应函数模型的关系，系统地研究有限秩酉扰动的谱理论。秩一的设置是合理的理解，而有限秩问题的一般处理提出了几个题外话。最初，非循环酉未扰动算子提出了一个问题。这个问题现在已经解决了。一般的问题涉及矩阵值Herglotz和柯西型变换。作为本项目的一部分，这些变换的研究将提供对有限秩微扰理论的深入了解。计划对所谓的外部柯西变换（如在秩一设置中所做的那样）进行正则化。进一步的调查将作出困难的情况下，无限秩扰动与缺陷运营商的迹类。还将研究类似的自伴设置以及微分算子的具体应用。

项目成果

Moment representations of exceptional X 1 orthogonal polynomials

异常 X 1 正交多项式的矩表示

• DOI: 10.1016/j.jmaa.2017.05.037
• 发表时间: 2017
• 期刊: Journal of Mathematical Analysis and Applications
• 影响因子: 1.3
• 作者: Kelly, Jessica Stewart;Liaw, Constanze;Osborn, John
• 通讯作者: Osborn, John

Boundary conditions associated with the general left-definite theory for differential operators

与微分算子的一般左定理论相关的边界条件

• DOI: 10.1016/j.jat.2018.10.005
• 发表时间: 2019
• 期刊: Journal of Approximation Theory
• 影响因子: 0.9
• 作者: Fleeman, Matthew;Frymark, Dale;Liaw, Constanze
• 通讯作者: Liaw, Constanze

Hyponormal Toeplitz operators with non-harmonic Symbol acting on the Bergman space

• DOI: 10.7153/oam-2019-13-04
• 发表时间: 2017-07
• 期刊: Operators and Matrices
• 影响因子: 0.5
• 作者: Matthew Fleeman;C. Liaw

Spectral approach to transport in a two-dimensional honeycomb lattice with substitutional disorder

• DOI: 10.1103/physrevb.99.024115
• 发表时间: 2017-11
• 期刊: Physical Review B
• 影响因子: 3.7
• 作者: E. Kostadinova;C. Liaw;A. Hering;A. Cameron;F. Guyton;L. Matthews;T. Hyde

Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems

最优多项式近似的零点：Jacobi 矩阵和 Jentzsch 型定理

• DOI: 10.4171/rmi/1064
• 发表时间: 2019
• 期刊: Revista Matemática Iberoamericana
• 作者: Bénéteau, Catherine;Khavinson, Dmitry;Liaw, Constanze;Seco, Daniel;Simanek, Brian
• 通讯作者: Simanek, Brian