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的角度研究量子/波力力学，及其在部分微分方程和几何学理论中的许多表现。当前的特定兴趣是在物理和几何环境中的散射共振，动力学和半经典的Zeta函数，量子混乱以及外部场中孤子/NLS的散射。正如流行的观点一样，共振是在某个频率下以最大振幅振荡的系统的趋势。从数学上讲，它用一个复杂的数字来描述，实际部分是频率和想象部分，衰减的速率（共振”为“钟声的垂死音符”）。这些数字显示为Meromorthic obletors或功能类别（例如Zeta函数，包括Riemann Zeta函数）。该项目着重于搜索共振分布中的一般数学原理，以及对此动机的特定示例的详细研究。先前的工作清楚地表明了这一趋势：共鸣出现在几何形状，半古典理论，障碍物散射，开放量子图中。有些结果普遍存在，有些结果在特定情况下是已知的。 我们对孤子散射的研究也是出于共振现象的动机，例如，在Bose-Einstein物质波的散射中寻找正确的共振传播概念。纯粹的数学量子图（经常与数字理论有联系的研究）用于对纳米结构进行建模和通过量子点传输。双曲线有理图的Zeta函数的零可以用作混沌散射中共振分布的模型。