Geometric Spectral Theory and Resonances

摘要：几何谱理论和共振(DMS-0901937)非紧致流形的谱理论主要是研究共振，共振类似于本征值，不同的是对应态通过色散到无穷大而衰变。几何背景下的共振研究是由几个基本问题驱动的：有多少个共振？它们是如何分配的？它们的分布告诉我们关于底层几何结构的什么信息？PI将研究在双曲空间上模拟的完整的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)在这些物体的内禀正则化不可用的情况下，开发行列式和迹公式等光谱工具；(3)逆散射问题--从共振集、散射相位等推断几何结构；(4)微扰问题--了解双曲度规扰动下共振的行为。这一方案的研究是由现代物理学的核心问题之一推动的，即了解物理系统的结构(例如其基本几何结构)与其对振荡刺激(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。眼睛和耳朵是振荡信号的接收器，大脑之所以能感知图像和声音，只是因为大脑具有非凡的解码能力。同样的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医学程序(如CAT扫描)，再到天文现象(如红移)。在所有这些情况下，观测数据都由振荡信号组成，需要进行数学分析来提取有关底层结构的信息。“谱论”是对相应数学领域的称谓。在抽象的数学背景下表达这些问题，强调了基本关系的普遍性，并促进了不同领域思想的交融。抽象光谱理论的进步已经并将继续导致在广泛的科学学科中的应用。PI已经写了一本书，并计划写其他书，这将有助于将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.