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Mathematical Theory of Resonances and Applications

共振数学理论及其应用

基本信息

批准号：
1800086
负责人：
Alexis Drouot
金额：
$ 15.29万
依托单位：
Columbia 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：
2018-07-01 至 2021-02-28
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800086&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一样，这些问题都有一个共同的特征：它们表现出围绕平衡状态呈指数衰减幅度的振荡。一个著名的理论例子涉及最近由 LIGO 探测器探测到的引力波。在技术环境中，振荡率和衰减率与平衡之间的比率称为品质因数。最小化该比率可以提高所涉及材料的稳定性和刚性特性。此类技术应用包括工程类石墨烯绝缘体；波导和光纤的概念；和高质量微电子系统的设计。对此类现象的数学研究依赖于称为共振理论的统一框架。它研究了能量可能逃逸或衰变情况下时谐态的自然概括。这些状态的广义能量形成一组离散的复数，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-鲁埃尔共振（在动力系统中）。它们的虚部描述了相关谐振态的典型频率，而实部则描述了它们在有界区域之外的衰减或增长。从数学上讲，它们是哈密顿解算子的亚纯延续的极点。大部分理论工作在于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。该项目专门研究周期性结构中对称破缺扰动下共振的出现；黑洞时空中霍金辐射的指数收敛；以及动力系统中的拓扑与 Pollicott-Ruelle 共振之间的关系。自然工具包括复分析和 Fredholm 算子理论；微局域框架内的经典到量子对应原理；该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。