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.

DMS-9970614 PI的主要兴趣是谐振和波方程的研究。通过复数描述的共振构成了在非伴随域上问题的特征值的替代，并且它们在数学和物理学的许多分支中自然而然地出现。共振的实际部分描述了状态的能量（或频率）以及其衰变速率的想象部分。与仅提供能量并假定状态永恒存在的特征值相比，这是一个更现实的模型。出现了新的现象，我们的理解仍然相当分散。线性和非线性波方程描述了许多物理现象通常与共振密切相关。在传播，光谱和散射理论的背景下，PI将进一步研究它们。 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混乱”在散射理论中。对于广泛的现象，可以通过Twoparameters来描述物理状态：其休息能和衰变速率。这些信息是在一个复杂数字中优雅编码的，其实际部分是能量和虚构部分，即衰减的速率。对一个称为共振的状态的描述在数学，物理和化学中自然出现：从Riemann Zeta函数到实验散射数据。PI研究了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细节中的行为。他还对“量子混乱”感兴趣，该研究在许多领域提出了问题：从数字理论到肠镜的物理系统。主要问题是世界的古典和量子观点之间的普遍关系 - 20世纪尚未解决的中央主题之一。