Many-Body Scattering

多体散射

• 批准号: 9970607
• 负责人: Maciej Zworski
• 金额: $ 8.06万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard 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=9970607&HistoricalAwards=false
• 关键词: Many; Body; Scattering

The object of the proposed research by Andras Vasy is the study of many-bodyscattering. The data from a scattering experiment are mathematicallyrepresented by the scattering matrices, or S-matrices. The central issuein many-body scattering is understanding the structure of these operators.A new method is now available for the analysis; it adopts techniques usedfor the study of hyperbolic equations. This project will employ thesemicrolocal techniques to analyze the wave front relation of the free-clusterto free-cluster scattering matrix in many-body scattering, involvingarbitrarily many particles, when the subsystems have no bound states.This S-matrix represents scattering data when both the incoming and theoutgoing particles are asymptotically free, i.e. they are not bound together.Its wave front relation describes the singularities of the outgoing datain terms of those of the incoming ones. More precisely, the main goal of thepresent project is to prove that if the subsystems of a many-body systemhave no bound states, e.g. if all potentials are non-negative, then the wavefront relation of the free-cluster to free-cluster S-matrix is given bythe broken geodesic flow to distance pi on the sphere. The absence of boundstates simplifies the phase space structure of the problem and henceprovides an ideal first step towards the understanding of the S-matricesin general. In addition, in this situation the problem is geometricallyanalogous to the wave equation in domains with corners. Thus, the completionof the project will demonstrate that scattering is in many respects ahyperbolic problem, similar to the wave equation. The next step will beto analyze what happens if there are bound states in some subsystems;in this case one has to combine the behavior of bound states with theclassical dynamics to understand the propagation of singularities.Indeed, many people are familiar with the following two descriptions ofthe propagation of light. First, in geometric optics, light propagatesin straight lines, reflecting from surfaces according to Snell's law.That is, the angles of incidence and of reflection are the same, as iflight consisted of little billiard balls. Second, light can be described bythe wave equation, its propagation thus being similar to that of waterwaves. There are many ways in this particular example in which the firstpicture gives a rough description of the second, but the followingconnection has particularly wide-ranging generalizations: singularities ofthe amplitude of the wave (`sharp signals' or `jumps in signals') propagatealong light rays. The purpose of the current proposal is to investigate theanalogous relationship between classical and quantum mechanical particles,that is, to analyze in what ways does the motion of classical particles,considered as little billiard balls colliding with each other, describethe motion of quantum mechanical particles. In more concrete terms, how welldoes the motion of four billiard balls describe how four electrons collidewith each other? Since it is easier to analyze the corresponding propertiesof classical systems than those of quantum ones, just as it is easier todescribe light moving in straight lines than to solve the wave equation,the answer to this question significantly improves our understanding ofquantum mechanics.

Andras Vasy拟议的研究的目的是对多生成的研究。来自散射实验的数据在数学上通过散射矩阵或S-矩阵代表了。多体散射中的核心问题是了解这些操作员的结构。现在可以使用一种新方法。它采用了用于研究双曲方程的技术。该项目将采用脑局限技术来分析自由群体与自由群体散射矩阵的波前关系，在多体散射中，涉及许多粒子，当子系统无界状态无绑定状态时，S-matrix代表了界面的散射，当散发性散布在偶数和界面上，这些粒子在无限的范围内都不是相关的。在即将到来的术语中的传入术语中。更确切地说，《祖先项目》的主要目标是证明，如果多体系的子系统无界状态，例如如果所有电势都是非负的，则自由群集与自由群集S-Matrix的波前关系由损坏的测地层流到球体上的距离PI。边界的缺失简化了问题的相空间结构，因此为理解S-摩西蛋白的一般迈出了理想的第一步。此外，在这种情况下，问题与有角落的域中的波动方程式是几何学的。因此，该项目的完成将证明散射在许多方面都是aberbolic问题，类似于波方程。下一步将分析如果某些子系统中有界面状态会发生什么；在这种情况下，必须将约束状态的行为与典型的动态结合起来，以了解奇异性的传播。事实上，许多人熟悉了光传播的两个描述。首先，在几何光学元件中，光线直线，根据Snell定律反射表面，即入射率和反射角度相同，就像由小台球球组成的Iflight。其次，波方程可以描述光，因此其传播与水波相似。在这个特定示例中，有很多方式对第二个图进行了粗略的描述，但是以下连接具有特别广泛的概括：波浪的振幅（“尖锐信号”或“信号中的跳跃”）propagateAlong灯光射线。当前建议的目的是研究经典机械颗粒和量子机械颗粒之间的造成关系，也就是说，分析经典颗粒的运动的方式，被认为是彼此相互冲突的小台球球，描述了量子机械颗粒的运动。用更具体的术语，四个台球球的运动如何很好地描述了四个电子如何相互碰撞？由于分析经典系统的相应属性比量子系统更容易，就像在直线上移动的光更容易，而不是解决波动方程，这个问题的答案显着提高了我们对Quantum力学的理解。