权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Mathematical Theory of Resonances and Applications

共振数学理论及其应用

基本信息

批准号：
2118608
负责人：
Alexis Drouot
金额：
$ 15.29万
依托单位：
University of Washington
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-02-01 至 2022-11-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2118608&HistoricalAwards=false
关键词：
Mathematical Theory Resonances Applications

项目摘要

The PI investigates various problems arising in quantum and classical scattering: condensed matter physics and the study of surface states; Hawking radiation in general relativity; chaotic dynamics in statistical systems. Like a bell sounding its last notes, these problems all share a common feature: they exhibit oscillations with exponentially decaying amplitudes around an equilibrium state. A famous theoretical example concerns gravitational waves, recently detected by the LIGO detector. In technological settings, the ratio between the oscillation and decay rates to equilibrium is called the quality factor. Minimizing this ratio improves the stability and rigidity properties of the materials involved. Such technological applications include engineering graphene-like insulators; conception of waveguides and optic fibers; and design of high-quality microelectrical systems. The mathematical study of such phenomena relies on a unified framework called resonance theory. It investigates natural generalizations of time-harmonic states in situations where the energy may escape or decay. Generalized energies of such states form a discrete set of complex numbers called quantum resonances (in quantum scattering), quasinormal modes (in general relativity) or Pollicott--Ruelle resonances (in dynamical systems). Their imaginary parts describe the typical frequency of the associated resonant states while the real parts describe their decay or growth outside bounded regions. Mathematically, they are the poles of the meromorphic continuation of the Hamiltonian resolvent. Much of the theoretical work consists in studying their stability and asymptotic distributions; and how they affect time evolution. This project specifically deals with the emergence of resonances under symmetry-breaking perturbations in periodic structures; the exponential convergence of Hawking radiation in black-hole spacetimes; and the relation between topology and Pollicott--Ruelle resonances in dynamical systems. Natural tools include complex analysis and Fredholm operator theory; the classical-to-quantum correspondence principle within the microlocal framework; and partial differential equations techniques such as multiscale analysis.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.

PI研究量子和经典散射中出现的各种问题：凝聚态物理和表面态研究;广义相对论中的霍金辐射;统计系统中的混沌动力学。就像钟声敲响最后的音符，这些问题都有一个共同的功能：它们在平衡态附近表现出振幅呈指数衰减的振荡。一个著名的理论例子是最近由LIGO探测器探测到的引力波。在技术设置中，振荡和衰减速率之间的比率平衡被称为品质因数。最小化该比率提高了所涉及的材料的稳定性和刚性特性。这些技术应用包括工程石墨烯类绝缘体;波导和光纤的概念;以及高质量微电子系统的设计。这种现象的数学研究依赖于一个统一的框架，称为共振理论。它调查自然概括的时间谐波状态的情况下，能量可能逃逸或衰减。这些状态的广义能量形成了一组离散的复数，称为量子共振（在量子散射中），准正规模式（在广义相对论中）或波利科特-吕尔共振（在动力学系统中）。它们的虚部描述了相关共振态的典型频率，而真实的部分描述了它们在有界区域之外的衰减或增长。在数学上，它们是哈密顿预解式的亚纯延拓的极点。大部分的理论工作包括研究它们的稳定性和渐近分布，以及它们如何影响时间演化。该项目具体涉及周期性结构中破环扰动下共振的出现;黑洞时空中霍金辐射的指数收敛;以及动力系统中拓扑结构与波利科特-吕勒共振之间的关系。自然的工具包括复分析和Fredholm算子理论;微局域框架内的经典到量子对应原理;以及偏微分方程技术，如多尺度分析。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。