Scattering Theory

散射理论

• 批准号: 0654436
• 负责人: Maciej Zworski
• 金额: $ 44.99万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0654436&HistoricalAwards=false
• 关键词: Scattering; Theory

The purpose of the project is the study of quantum/wave mechanics from the mathematical point of view, and of its many manifestations in the theory of partial differential equations and geometry. Specific current interests are the distribution of scattering resonances in physical and geometric settings, dynamical and semiclassical zeta functions, quantum chaos, and scattering of solitons/NLS in external fields.As popular view would have it, resonance is the tendency of a system to oscillate at a maximum amplitude at a certain frequency. Mathematically, it is described by a complex number with the real part being the frequency and the imaginay part, the rate of decay (the resonances "die" as "dying notes of a bell"). These numbers appear as poles of classes of meromorphic operators or functions (such as zeta functions, including the Riemann zeta function).The project focuses on the search for general mathematical principles in the distribution of resonances, and on the detailed study of specific examples motivated by that. The previous work clearly demonstrates this trend: resonances appear in geometry, semi-classical theories, obstacle scattering, open quantum maps.Some results hold universally and some are known in specific cases. Our study of scattering of solitons is also motivated by resonance phenomena, such as the search for the correct concept of resonance transmission in scattering of Bose-Einstein matter waves.The phenomena studied in the project are very general: for instance, microwaves can be used to model quantum scattering and quantum chaos, leading to insights about MEMS (micro-electro-nechanical systems) which are constructed using tiny resonators. Purely mathematical quantum maps (the study of which often has connections to number theory) are used to model nanostructures and transport through quantum dots. Zeros of zeta functions for hyperbolic rational maps can be used as models for resonance distribution in chaotic scattering.

该项目的目的是从数学角度研究量子/波力学及其在偏微分方程和几何理论中的多种表现形式。当前的具体兴趣是物理和几何环境中散射共振的分布、动力学和半经典 zeta 函数、量子混沌以及外部场中孤子/NLS 的散射。正如流行的观点所认为的，共振是系统振荡的趋势在某个频率下达到最大振幅。在数学上，它由一个复数来描述，实部是频率，虚部是衰减率（共振“消亡”为“钟声的垂死音符”）。这些数字显示为亚纯算子或函数（例如 zeta 函数，包括黎曼 zeta 函数）类别的极点。该项目侧重于寻找共振分布中的一般数学原理，以及对具体示例的详细研究就这样。之前的工作清楚地证明了这一趋势：共振出现在几何、半经典理论、障碍物散射、开放量子图谱中。有些结果是普遍适用的，有些结果是在特定情况下已知的。 我们对孤子散射的研究也是由共振现象推动的，例如寻找玻色-爱因斯坦物质波散射中共振传输的正确概念。该项目中研究的现象非常普遍：例如，可以使用微波模拟量子散射和量子混沌，从而深入了解使用微型谐振器构建的 MEMS（微机电系统）。纯数学量子图（其研究通常与数论有关）用于模拟纳米结构和通过量子点的传输。双曲有理图的 zeta 函数零点可用作混沌散射中共振分布的模型。