Propagation Phenomena for Waves and Scattering

所提出的研究的对象是在不同的情况下的传播现象的研究。该提案中的主题涵盖了各种不同的领域，如边缘波动方程的衍射现象的研究，以及弯曲时空中波的渐近行为，如广义相对论。该项目将采用几何微局部或相空间技术来探索这些领域的新问题。这些技术中的一些是建设性的，例如在一类渐近Anti-de Sitter（AdS）空间上提出的前向基本解和散射算子的构造，而另一些则依赖于相空间能量估计，如微局部正交换子估计，预计将用于研究弯曲时空上波动方程的渐近行为（如克尔德西特空间，描述旋转黑洞），波动方程的整体条件取代初始条件，并在波传播的弯曲时空与渐近边缘和尖点。甚至对应于平稳（无传播）问题的椭圆问题有时也可以从不同的观点受益：关于渐近双曲空间上拉普拉斯算子的预解式的亚纯延拓及其高能量行为的最近工作提供了一个例子。波传播在物理世界中无处不在，电磁波（例如，光波）是最普遍的例子之一。一个典型的问题是波如何与边缘等结构相互作用，推广了众所周知的镜子反射定律。散射理论是另一个受传播现象支配的学科。例如，散射现象进入大尺度量子波的描述，数学模型为“无穷大”。“由于传播现象在物理世界中无处不在，因此对其更精确的理解具有许多潜在的应用。预期感兴趣的具体物理应用是发展散射理论（即，研究光波或量子粒子在弯曲时空中的全局行为，在广义相对论的框架内描述我们的宇宙）。所涉及的一些技术预计也将有利于理解成像问题，例如，在地球物理方面。

The object of the proposed research is the study of propagation phenomena in different contexts. Topics in the proposal encompass such diverse areas as the study of diffractive phenomena for the wave equation at edges, as well as the asymptotic behavior of waves on curved space-times, such as in general relativity. The project will employ geometric microlocal, or phase space, techniques to explore new problems in these areas. Some of these techniques are constructive, such as the proposed construction of the forward fundamental solution and the scattering operator on a class of asymptotically Anti-de Sitter (AdS) spaces, while others rely on phase-space energy estimates, such as the microlocal positive commutator estimates that are expected to be of use in the study of the asymptotic behavior of the wave equation on curved space-times (such as Kerr-de Sitter space, describing rotating black holes), of wave equations with global conditions replacing initial conditions, and in wave propagation on curved space-times with asymptotic edges and cusps. Even elliptic problems, which correspond to stationary (no propagation) problems, can sometimes benefit from a different point of view: recent work on the meromorphic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces, and on its high energy behavior, provides an example.Wave propagation is ubiquitous in the physical world, with electromagnetic waves (e.g., light waves) being one of the most prevalent examples. A typical question is how waves interact with structures such as edges, generalizing the well-known law of reflection from mirrors. Scattering theory is another subject governed by propagation phenomena. For instance, scattering phenomena enter into the description of quantum waves on large scales, mathematically modelled as "infinity." As propagation phenomena are ubiquitous in the physical world, their more precise understanding has many potential applications. A concrete physical application that is expected to be of interest is developing scattering theory (i.e., investigating how light waves or quantum particles behave globally, in the context of curved space-times, which give a description of our universe in the framework of general relativity). Some of the techniques involved are also expected to benefit the understanding of imaging problems, for instance, in the geophysical context.