Scattering Theory and Semi-Classical Analysis

散射理论是在比相互作用区域本身大得多的时间和/或距离尺度上研究相互作用系统。散射实验在自然科学中被广泛用于确定遥远物体的结构，如恒星，非常小的物体，如原子，或难以到达的物体，如地心。例如，在1909年的一次著名实验中，通过研究阿尔法粒子通过金箔的散射模式，卢瑟福确定原子是由一个被电子包围的原子核组成的。半经典分析是相空间分析的一种，它研究算子和相关对象对一个小参数的依赖关系。半经典分析最初是受玻尔或量子经典对应原理的启发，该原理断言经典力学是量子力学的极限，因为普朗克常数趋于0。然而，半经典分析在不同的科学领域发现了许多应用。小参数在这些场中的作用可以通过Born-Oppenheimer近似中的核质量平方根的倒数、固体物理中的磁场强度、绝热理论中的绝热参数、高能谱问题中能量的平方根的倒数等来发挥。通过建立在半经典分析和散射理论的最新进展的基础上，这笔赠款支持的研究旨在为这些领域的以下方面做出贡献。首先，它试图通过将量子散射物体与半经典极限中的经典物体联系起来来建立量子-经典对应的新实例。第二，它的目的是主要研究粒子或信号存在时的散射，这些粒子或信号永远不会离开相互作用区域。到目前为止，这种构型被认为对散射系统的行为有重大影响，但很少有文献完全阐明这些构型。最后，为了达到这些目的，本研究将进一步发展半经典傅立叶积分算子的理论。这些算子的有趣而重要的性质使它们不仅成为半经典分析的有力工具，而且在相关领域也是如此。所有这些发展都有可能有助于反问题的解决以及上述科学领域的进步。

Scattering theory is the study of interacting systems on scales of time and/or distance much larger than those of the interaction region itself. Scattering experiments are widely used in the natural sciences to determine the structure of objects which are far away, like the stars, very small, like the atom, or difficult to reach, like the Earth's core. In a famous experiment of 1909, for example, by studying the scattering pattern of alpha particles passing through gold foil Rutherford determined that the atom consisted of a nucleus surrounded by electrons. Semi-classical analysis is a type of phase space analysis which investigates the dependence of operators and related objects on a small parameter. Originally motivated by Bohr's, or the quantum-classical, correspondence principle, which asserts that classical mechanics is the limit of quantum mechanics as Planck's constant tends to 0, semi-classical analysis has found, however, many applications in diverse areas of science. The role of the small parameter in these fields can be played by the inverse of the square root of the nuclear mass in the Born-Oppenheimer approximation, the magnetic field strength in solid-state physics, the adiabatic parameter in adiabatic theory, the inverse of the square root of the energy in high-energy spectral problems, and others. By building upon recent advances in semi-classical analysis and scattering theory the research supported by this grant aims to contribute to the following aspects of these fields. First, it seeks to establish new instances of the quantum-classical correspondence by relating quantum scattering objects to classical ones in the semi-classical limit. Second, it aims to investigate mainly scattering in the presence of particles or signals which never leave the interaction region. Such configurations are by now recognized to have a significant impact on the behavior of scattering systems but very few of them have been completed elucidated in the literature. Lastly, to achieve these goals this research will further the theory of semi-classical Fourier integral operators. The interesting and important properties of these operators have made them a strong tool not only in semi-classical analysis but also in related fields. All of these developments have the potential to contribute to the solutions of inverse problems as well as to the advancement of the fields of science mentioned above.