Scattering Theory

该项目的目的是从数学的角度研究量子/波动力学，以及它在偏微分方程理论和几何中的许多表现。目前的具体兴趣是物理和几何设置中的散射共振分布，动态和半经典zeta函数，量子混沌，以及外场中孤子/NLS的散射。正如流行的观点所认为的那样，共振是系统在某一频率上以最大振幅振荡的趋势。在数学上，它是用一个复数来描述的，其中实部是频率，虚部是衰减率（共振“死亡”就像“钟的垂死音符”）。这些数作为亚纯算子或函数类（如ζ函数，包括黎曼ζ函数）的极点出现。该项目的重点是寻找共振分布的一般数学原理，并对由此产生的具体例子进行详细研究。先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、开放量子映射中。有些结果是普遍适用的，有些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激发的，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。该项目研究的现象非常普遍：例如，微波可以用来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微机电系统（MEMS）。纯数学量子图（对它的研究通常与数论有关）被用来模拟纳米结构和通过量子点的传输。双曲有理映射的ζ函数零点可以作为混沌散射中共振分布的模型。

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.