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Asymmetry, Embedded Eigenvalues, and Resonance for Differential Operators

微分算子的不对称性、嵌入特征值和共振

基本信息

批准号：
1814902
负责人：
Stephen Shipman
金额：
$ 24.5万
依托单位：
Louisiana State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1814902&HistoricalAwards=false
关键词：
Asymmetry Embedded Eigenvalues Resonance Differential

项目摘要

This project addresses the mathematical foundations of physical principles that underlie the design of optical and electronic devices. The pervading concept is the principle of "resonance", which lies behind lasers, filters, antennas, and light-emitting diodes, etc. The technical term "embedded eigenvalue" in the title refers to a mathematical concept that is intimately intertwined with resonance; it is related to specific configurations of electromagnetic fields in a structure or device and how they interact with external inputs. The project combines theory with computer simulations. The principal investigator runs a research seminar, which integrates seasoned researchers, a post-doctoral associate, doctoral students, and undergraduate students, each of whom is involved in a specific aspect of the project. Emphasis is placed on developing methods at the interfaces of mathematics, physics, and engineering, paying special attention to bridging communication gaps between the disciplines and training a new generation of researchers who can work in an interdisciplinary setting.Classically familiar examples of embedded eigenvalues are induced by finite symmetry groups of a differential operator. Recent demonstrations of non-symmetry-induced spectrally embedded bound states in electromagnetic structures have opened the way toward a richer set of resonance phenomena. This project develops a theory of asymmetry and embedded eigenvalues. The investigations employ functional analysis and partial differential equations, complex analysis, spectral theory, and analytic geometry. These are the specific problems of the project: (1) It investigates the precise connection between asymmetry, embedded eigenvalues, and reducibility of an analytic variety called the "Fermi surface" for periodic operators, including the partial differential equations (PDE) of physical systems and mathematical graph models; (2) The principal investigator's recent work shows that bilayer quantum-graph graphene is special in that its Fermi surface is always reducible, regardless of the type of asymmetries in the potentials on the edges connecting the two sheets of graphene. The project seeks the mathematical reasons behind this and the PDE counterpart; (3) The investigator and collaborators are developing numerical algorithms for probing the very complex situation of embedded eigenvalues for the Maxwell equations of electromagnetics; (4) The existence and construction of embedded eigenvalues for the Neumann-Poincare boundary-integral operator that is fundamental to the theory of "plasmonic resonances" is also being studied.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目涉及光学和电子设备设计的物理原理的数学基础。这个普遍的概念是“谐振”的原则，它是激光器，滤波器，天线和发光二极管等的背后。标题中的技术术语“嵌入本征值”是指一个与谐振密切相关的数学概念;它与结构或设备中电磁场的特定配置以及它们如何与外部输入相互作用有关。该项目将理论与计算机模拟相结合。首席研究员主持一个研究研讨会，其中包括经验丰富的研究人员，博士后助理，博士生和本科生，每个人都参与了该项目的一个特定方面。重点放在开发方法的接口的数学，物理和工程，特别注意弥合学科之间的沟通差距，培养新一代的研究人员谁可以在一个跨学科的setting.Classically熟悉的例子嵌入特征值是由有限对称群的微分算子诱导。最近的非对称性诱导的光谱嵌入束缚态的电磁结构的演示开辟了一条道路，走向更丰富的一套共振现象。本计画发展不对称性与嵌入本征值的理论。研究采用泛函分析和偏微分方程、复分析、谱理论和解析几何。本项目的具体问题是：（1）研究周期算子的非对称性、嵌入特征值和称为“费米面”的解析变量的约化之间的精确联系，包括物理系统和数学图形模型的偏微分方程（PDE）;（2）主要研究者最近的工作表明，双层量子图形石墨烯的特殊之处在于它的费米面总是可还原的，而不管连接两个石墨烯片的边缘上的电势的不对称性的类型。（3）研究人员和合作者正在开发数值算法，以探测电磁学麦克斯韦方程组嵌入本征值的非常复杂的情况;（4）Neumann-Poincare边界积分算子的嵌入本征值的存在性和构造，这是“等离子体共振”理论的基础。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。