Many-Body Scattering

Andras Vasy提出的研究目标是研究多体散射。散射实验的数据用散射矩阵或S矩阵表示。多体散射的中心问题是理解这些算符的结构。现在有一种新的分析方法，它采用了研究双曲方程的技术。本计画将利用微局域技术分析多体散射中自由团簇与自由团簇散射矩阵的波前关系，当子系统没有束缚态时，此S矩阵表示入射和出射粒子都渐近自由时的散射数据，也就是说，它们没有被束缚在一起。它的波前关系描述了输出数据的奇异性与输入数据的奇异性的关系。更确切地说，本项目的主要目标是证明，如果一个多体系统的子系统没有束缚态，例如，如果所有的势都是非负的，那么自由团簇与自由团簇S-矩阵的波前关系由球面上距离π的破测地线流给出。没有束缚态简化了问题的相空间结构，从而为理解S-矩阵提供了理想的第一步。此外，在这种情况下，该问题在几何上类似于具有角点的域中的波动方程。因此，该项目的完成将证明散射在许多方面是一个双曲问题，类似于波动方程。下一步将是分析在某些子系统中存在束缚态时会发生什么情况;在这种情况下，我们必须将束缚态的行为与经典动力学联合收割机结合起来，以理解奇点的传播。实际上，许多人都熟悉以下两种对光传播的描述。首先，在几何光学中，光沿直线传播，并根据斯涅耳定律从表面反射，也就是说，入射角和反射角是相同的，就像光是由小台球组成的一样。其次，光可以用波动方程来描述，因此它的传播与水波的传播相似。在这个特殊的例子中，第一幅图对第二幅图有许多粗略的描述，但下面的联系具有特别广泛的概括性：波的振幅的奇异性（“尖锐信号”或“信号跳跃”）沿着光线传播。本文的目的是研究经典粒子和量子粒子之间的相似关系，即分析经典粒子的运动（被认为是相互碰撞的小台球）如何描述量子粒子的运动。更具体地说，四个台球的运动如何描述四个电子如何相互碰撞？由于分析经典系统的相应性质比分析量子系统的相应性质更容易，就像描述直线运动的光比求解波动方程更容易一样，这个问题的答案大大提高了我们对量子力学的理解。

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.