Scattering Theory

散射理论

• 批准号: 9970614
• 负责人: Maciej Zworski
• 金额: $ 13.5万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970614&HistoricalAwards=false
• 关键词: Scattering; Theory

DMS-9970614The main interest of the PI is the study of resonances and of the wave equation. Resonances which are described by complex numbers constitute a replacement of eigenvalues for problems on noncompact domains and they appear naturally in many branches of mathematics and physics. The real part of a resonance describes the energy (or frequency) of a state and the imaginary part its rate of decay. This constitutes a more realistic model than an eigenvalue which provides energy only and assumes eternal existence of a state. A new wealth of phenomena appears and our understanding is still rather fragmentary. Linear and non-linear wave equations describe many physical phenomena often closely related to resonances. In the contexts of propagation, spectral and scattering theories the PI will investigate them further. The specific directions include: understanding of the dynamical definition of resonances and their appearance in the long time behaviour of solutions of the wave and Schroedinger equations, obtaining lower bounds on the number of resonances in terms of dynamical quantities, developing the theory of the FBI transformation on non-compact manifolds with applications to resonances and to the study of propagation for Schroedinger equation in mind, estimates of resonances at low energies, understanding of "quantum chaos" in scattering theory.For a broad range of phenomena, a physical state can be described by twoparameters: its rest energy and its rate of decay. This information is elegantly encoded in a complex number, whose real part is the energy and the imaginary part, the rate of decay. This description of a state, called a resonance, appears naturally in mathematics, physics and chemistry: from the Riemann zeta function to experimental scattering data.The PI studies general principles in the distribution of resonances,their relation to wave propagation, and their behaviour in various specificsituations. He is also interested in "quantum chaos", the study of which raises questions in many areas: from number theory to mezoscopic systems of physics. The main issues are universal relations between the classical and quantum views of the world -- one of the yet unresolved central themes of the 20th century.

PI的主要兴趣是共振和波动方程的研究。用复数描述的共振构成了非紧域问题的特征值的一种替代，它们在数学和物理的许多分支中自然出现。共振的实部描述状态的能量（或频率），虚部描述状态的衰减速率。这构成了一个比只提供能量并假设状态永恒存在的特征值更现实的模型。新的丰富的现象出现了，而我们的认识还相当零碎。线性和非线性波动方程描述了许多与共振密切相关的物理现象。在传播，光谱和散射理论的背景下，PI将进一步研究它们。具体方向包括：理解共振的动力学定义及其在波和薛定谔方程解的长时间行为中的表现，从动力学量的角度获得共振数的下界，发展非紧流形上的FBI变换理论，并将其应用于共振和薛定谔方程的传播研究，低能共振的估计，对散射理论中“量子混沌”的理解。对于广泛的现象，物理状态可以用两个参数来描述：它的静止能量和衰变速率。这个信息被优雅地编码成一个复数，它的实部是能量，虚部是衰减速率。这种状态的描述，称为共振，自然出现在数学、物理和化学中：从黎曼ζ函数到实验散射数据。PI研究共振分布的一般原理，它们与波传播的关系，以及它们在各种特定情况下的行为。他对“量子混沌”也很感兴趣，对它的研究在许多领域提出了问题：从数论到物理学的宏观系统。主要问题是经典世界观和量子世界观之间的普遍关系——这是20世纪尚未解决的中心主题之一。