Scattering Theory

DMS-9970614 PI的主要兴趣是研究共振和波动方程。用复数描述的共振是非紧域上问题特征值的一种替代，自然地出现在数学和物理的许多分支中。共振的真实的部分描述了一个状态的能量（或频率），虚部描述了它的衰减速率。这构成了一个比本征值更现实的模型，本征值只提供能量，并假设一个状态的永恒存在。大量新的现象出现了，而我们的理解仍然相当零碎。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出现，获得动力学量方面共振数的下限，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，估计低能量下的共振，对散射理论中的“量子混沌”的理解。对于广泛的现象，一个物理状态可以用两个参数来描述：它的静止能量和它的衰变率。这些信息被巧妙地编码成一个复数，其真实的部分是能量，虚部是衰变率。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zeta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。他还对“量子混沌”感兴趣，对它的研究在许多领域提出了问题：从数论到物理学的中观系统。主要问题是世界的经典和量子观点之间的普遍关系--这是世纪尚未解决的中心主题之一。

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.