Asymmetry, Embedded Eigenvalues, and Resonance for Differential Operators

该项目解决了光学和电子设备设计背后的物理原理的数学基础。普遍的概念是“共振”原理，它隐藏在激光器、滤波器、天线和发光二极管等背后。标题中的技术术语“嵌入特征值”指的是与共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，由经验丰富的研究人员、博士后助理、博士生和本科生组成，每个人都参与项目的一个特定方面。重点放在发展方法在数学，物理和工程的接口，特别注意弥合学科之间的沟通差距和培训新一代的研究人员谁可以在跨学科的设置工作。经典上熟悉的嵌入特征值的例子是由微分算子的有限对称群导出的。最近在电磁结构中非对称诱导的频谱嵌入束缚态的证明为更丰富的共振现象开辟了道路。本项目发展了不对称和嵌入特征值的理论。研究采用泛函分析和偏微分方程、复分析、光谱理论和解析几何。这些是该项目的具体问题：(1)研究周期算子的不对称性、嵌入特征值和解析型“费米曲面”的可约性之间的精确联系，包括物理系统和数学图模型的偏微分方程（PDE）；(2)首席研究员最近的工作表明，双层量子图石墨烯的特殊之处在于，它的费米面总是可约的，无论连接两片石墨烯的边缘上的势的不对称类型如何。该项目寻求背后的数学原因和PDE对应；(3)研究者和合作者正在开发数值算法，用于探测电磁学麦克斯韦方程嵌入特征值的非常复杂的情况；(4) Neumann-Poincare边界积分算子的嵌入特征值的存在性和构造也是“等离子体共振”理论的基础。该奖项反映了美国国家科学基金会的法定使命，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。

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.