Geometric Spectral Theory and Resonances

摘要：几何谱理论和共振（DMS-0901937）非紧流形的谱理论主要是研究共振，共振类似于本征值，除了相应的状态会因色散而衰减到无穷大。 在几何背景下研究共振是由几个基本问题驱动的：有多少共振？它们是如何分布的？它们的分布告诉了我们什么关于潜在的几何形状？ PI将研究以双曲空间为模型的完整无限体积黎曼流形的谱几何。 这些空间是混沌量子散射的自然模型。 该项目的具体研究目标包括：（1）研究共振计数函数，并将共形极点的贡献与真实共振分离，（2）在这些物体的内在正则化不可用的情况下开发谱工具，如行列式和迹公式，（3）逆散射问题-从共振集，散射相位等推导几何结构，(4)微扰问题-了解双曲度规微扰下共振的行为。这项研究的动机是现代物理学的核心问题之一，即理解物理系统的结构（例如其基本几何结构）与其对振荡刺激（例如光波或声波）的响应之间的关系。 也许这种关系最基本的例子是人类的视觉和听觉。 眼睛和耳朵是振动信号的接收器，大脑之所以能感知图像和声音，是因为大脑具有非凡的解码能力。 同样基本关系的其他例子比比皆是，从涉及粒子束碰撞的现代物理实验到CAT扫描等医疗程序，再到红移等天文现象。 在所有这些情况下，观测数据由振荡信号组成，需要进行数学分析以提取有关底层结构的信息。 “谱理论”是数学中相应领域的名称。 在抽象的数学环境中表达这些问题，强调了基本关系的普遍性，并促进了来自不同领域的思想的交叉。 抽象光谱理论的进步已经并将继续导致广泛的科学学科的应用。 PI已经写了一本书，并计划写其他书，这将有助于使这项研究的结果提供给广泛的研究受众，包括物理学家和数学家。 PI目前有两名博士生。

Abstract: Geometric Spectral Theory and Resonances (DMS-0901937) The spectral theory of non-compact manifolds is dominated by the study of resonances, which are analogous to eigenvalues except that the corresponding states are subject to decay by dispersion to infinity. The study of resonances in a geometric context is driven by a few basic questions: How many resonances are there? How are they distributed? What does their distribution tell us about the underlying geometry? The PI will study the spectral geometry of complete, infinite-volume Riemannian manifolds modeled on hyperbolic spaces. These spaces are natural models of chaotic quantum scattering. The specific research goals of this project include: (1) Studying the resonance counting function and separating contributions from conformal poles from the true resonances, (2) developing spectral tools such as determinants and trace formulas in cases where intrinsic regularizations of these objects are not available, (3) inverse scattering problems - deducing geometric structure from the resonance set, scattering phase, etc., (4) perturbation problems - understanding the behavior of resonances under perturbations of a hyperbolic metric. The research in this proposal is motivated by one of the core issues in modern physics, which is to understand the relationship between the structure of a physical system, e.g. its underlying geometry, and its response to oscillatory stimulus such as light or sound waves. Perhaps the most fundamental examples of this relationship are the human senses of vision and hearing. Eyes and ears are receptors for oscillatory signals, and the mind perceives images and sounds only because of the brain's remarkable ability to decode them. Other cases of the same basic relationship abound, from modern physics experiments involving particle beam collision to medical procedures such as the CAT scan to astronomical phenomena such as red shift. In all of these situations the observational data consist of oscillatory signals, and mathematical analysis is required to extract information about the underlying structure. "Spectral theory" is the title given to the corresponding field of mathematics. Expressing these problems in an abstract mathematical setting emphasizes the universality of the fundamental relationship and fosters the cross-fertilization of ideas from different areas. Advances in abstract spectral theory have led and will continue to lead to applications across a broad range of scientific disciplines. The PI has written one book and has plans to write others, which will help make the results of this research available to a broad research audience, including both physicists and mathematicians. The PI currently has two doctoral students.