Many-body Scattering and Symmetric Spaces

摘要：提出的研究对象是散射理论的几何研究。这包含了诸如多体散射和对称空间上的拉普拉斯等明显遥远的领域。最近从双曲方程的研究中获得的强大几何技术已经在多体散射的详细研究中发挥了重要作用，多体散射描述了量子粒子如何相互作用，例如在加速器中。该项目将利用这些微局部技术来探索多体散射中的新问题，并为对称空间的分析带来见解。这些问题包括在阈值能量下粒子行为的详细研究，阈值能量是粒子的新构型可以接近的能级。对于许多化学现象，普朗克常数h可能被认为很小，这激发了对半经典多体现象的研究。一个特别的例子是谱移函数，这是PI与王雪平正在进行的一个联合项目的主题。与Gunther Uhlmann联合提出的另一个研究领域是有限能量逆多体散射，即是否可以从散射矩阵确定粒子之间的相互作用，散射矩阵是描述散射实验结果的对象。也许令人惊讶的是，有一些非常代数的对象类似于多体问题，即高秩对称空间。这些空间中所谓的平面的结构与多体散射中构型空间的结构非常相似，Weyl室的壁起到了碰撞面的作用。在一个联合项目中，Rafe Mazzeo和PI计划使用多体散射的构造技术来获得这些空间上各种解析对象的完全渐近行为，表明在那里观察到的许多现象具有更大的普遍性。的确，许多人都熟悉光的传播的以下两种描述。首先，在几何光学中，光沿直线传播，根据斯涅尔定律从表面反射。也就是说，入射角和反射角是相同的，就好像光是由小台球组成的。第二，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。这两种观点之间有密切的关系。也就是说，对于波动方程的解，用于携带信息的尖锐信号（或信号的“奇点”）的传播可以用更简单的几何光学图像来精确描述。在多粒子相互作用中，经典力学和量子力学之间也有类似的对应关系。也就是说，对于量子粒子，波函数在“无穷远处”的演化几乎可以用经典图像来描述，正如之前PI所证明的那样。经典图像和量子图像之间唯一的分歧是束缚态的出现。例如，对于一个由两个质子和两个电子组成的系统，一个质子和一个电子可以形成一个氢原子——如果一个高能电子击中它，氢原子可能会分裂。在这个提案中，PI将部分使用这些结果和相关工具，进一步研究多体现象，例如更精确地检查小普朗克常数的量子-经典对应关系，并分析是否可以从散射实验的结果确定粒子之间的相互作用，以及部分将这些结果扩展到更多的几何设置。因此，这些结果与物理和化学中的重要问题有关。

PI: Andras Vasy, MITDMS-0201092Abstract:The object of the proposed research is the geometric study of scattering theory. This encompasses such apparently distant areas as many-body scattering and the Laplacian on symmetric spaces. Powerful geometric techniques recently adapted from the study of hyperbolicequations have already played a major role in the detailed study of many-body scattering, which describes how quantum particles interact, e.g. in an accelerator. The proposed project will employ these microlocal techniques to explore new problems in many-body scattering, and to bringinsights to analysis on symmetric spaces. These problems include the detailed study of the behavior of particles at threshold energies, which are energy levels where new configurations of particles become accessible. For many chemical phenomena Planck's constant, h, may be considered small, motivating the study of semiclassical many-body phenomena. A particular example is the spectral shift function, which has been the subject of an ongoing joint project of the PI with Xue Ping Wang. Another area of proposed research, joint with Gunther Uhlmann, is finite energy inverse many-body scattering, i.e. whether one can determine the interaction between particles from the scattering matrices, which are objects describing the outcome of scattering experiments. Perhaps surprisingly, there are very algebraic objects which are analogous to many-body problems, namely higher rank symmetric spaces. The structureof so-called flats in these spaces is very similar to the structure of the configuration space in many-body scattering, with the walls of the Weyl chambers playing the role of collision planes. In a joint project, Rafe Mazzeo and the PI plan to use the constructive techniques from many-bodyscattering to obtain the full asymptotic behavior of various analytic objects on these spaces, showing that many of the phenomena observed there have their counterparts in much greater generality.Indeed, many people are familiar with the following two descriptions of the propagation of light. First, in geometric optics, light propagates in straight lines, reflecting from surfaces according to Snell's law. That is, the angles of incidence and of reflection are the same, as if lightconsisted of little billiard balls. Second, light can be described by the wave equation, its propagation thus being similar to that of water waves. There is a close relationship between these two viewpoints. Namely, for solutions of the wave equation, the propagation of sharp signals (or `singularities' of signals), which are the signals used to carry information, is precisely described by the simpler geometric optics picture. There is a similar correspondence between classical and quantum mechanics in many-particle interactions. Namely, for quantum particles theevolution of wave functions `at infinity' is almost described by the classical picture, as was proved previously by the PI. The only divergence between the classical and the quantum pictures is the emergence of bound states. For example, for a system consisting of two protons and twoelectrons, one of the protons and one of the electrons can form a hydrogen atom -- which may in turn break up if an electron with high energy hits it. In this proposal the PI will in part use these results, and related tools, for further investigation of many-body phenomena, such as examining the quantum-classical correspondence more precisely for small Planck's constant, and analyzing whether one can determine the interactions between particles from the result of a scattering experiment, and in part to extend such results to more geometric settings. These results are thus related to important questions in physics and chemistry.