Mathematical Theory of Resonances and Applications

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的研究；广义相对论中的霍金辐射；统计系统中的混沌动力学。就像敲响最后一个音符的钟一样，这些问题都有一个共同的特征：它们在平衡状态周围表现出指数衰减幅度的振荡。最近由LIGO探测器探测到的引力波就是一个著名的理论例子。在技术设置中，振荡率和衰减率达到平衡的比率称为质量因子。最小化这一比率可以提高所涉及材料的稳定性和刚性性能。这类技术应用包括工程类石墨烯绝缘体；波导和光纤的概念；并设计高质量的微电子系统。对这种现象的数学研究依赖于一个称为共振理论的统一框架。它研究在能量可能逃逸或衰减的情况下时调和态的自然推广。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），准正态模式（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共振态的典型频率，而实部描述了它们在有界区域外的衰减或增长。在数学上，它们是哈密顿解的亚纯延拓的极点。许多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变。本项目专门研究周期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；以及动力系统中拓扑与波利科特-鲁埃尔共振之间的关系。自然工具包括复分析和Fredholm算子理论；微局域框架中的经典-量子对应原理偏微分方程技术，比如多尺度分析。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。

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.