Wave propagation: singularities and asymptotics

拟议研究的对象是波的传播研究。这包含了不同的领域，如研究波动方程解的奇异性，这是波的“附近”行为，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技术来探索这些领域的新问题。其中一些技术是建设性的，如作者在描述渐近de-Sitter-like空间上的散射算子时所使用的技术，还有一些依赖于相空间能量估计，例如作者在证明奇点在有角的区域上的传播时所采用的微局部正交换子估计，以及在表明波被光滑边缘衍射时更规则时所采用的微局部正交换子估计。在适当的假设下，光的传播比入射波弱（可以解释为“弱”），这是与梅尔罗斯和翁施共同工作的结果。首先，在几何光学中，光以直线传播，根据斯涅耳定律（即，入射角和反射角是相同的，就好像光是由台球组成的一样）。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。这两种观点之间有着密切的联系。也就是说，对于波动方程的解，尖锐信号的传播（或信号的“奇点”，例如，通过瞬时打开光而获得）由更简单的几何光学图像精确地描述。这个项目的一部分可以被看作是这项工作的延伸到一个更一般的设置，如从弯曲的边缘反射。由于波的传播在物理世界中无处不在，因此对它的更精确理解具有许多潜在的应用，例如逆问题。在内部有裂缝的材料中，人们希望通过用波（例如声波或X射线）探测材料来找出这些裂缝的位置。这些裂纹通常是不光滑的（例如，它们可能有边或角），但它们仍然适合我们的一般几何框架。人们可以使用我们提出的结果-即，来自裂缝尖端的反射比来自光滑部分的反射弱（在某种意义上）-结合对来自光滑部分的反射的非常精确的理解，来定位裂缝。粗略的结论是：这些提示可以忽略不计。

The object of the proposed research is the study of wave propagation. This encompasses such diverse areas as the study of singularities of solutions of the wave equation, which is the "nearby" behavior of waves, as well as the asymptotic behavior of waves on curved space-times, such as in general relativity. The project will employ geometric microlocal techniques to explore new problems in these areas. Some of these techniques are constructive, such as the ones used by the author in the description of the scattering operator on asymptotically de-Sitter-like spaces, and some rely on phase-space energy estimates, such as the microlocal positive commutator estimates employed by the author in the proof of the propagation of singularities on domains with corners and in showing that the wave diffracted by a smooth edge is more regular (which can be interpreted as "weaker") than the incident wave under suitable hypotheses, in joint work with Melrose and Wunsch.Most people are familiar with the following two descriptions of the propagation of light. First, in geometric optics, light propagates in straight lines, reflecting from smooth surfaces according to Snell's law (i.e., the angles of incidence and of reflection are the same, as if light consisted of 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 obtained, for example, by turning on light instantaneously) is precisely described by the simpler geometric optics picture. Part of this project can be regarded as an extension of this work to a more general setting, such as reflections from curved edges. As wave propagation is ubiquitous in the physical world, a more precise understanding of it has many potential applications, for instance, to inverse problems. In a material with cracks inside it, one would like to find out the location of these cracks by probing the material with waves (say, sound waves, or x-rays). These cracks are typically not smooth (e.g., they may have edges or corners), but they still fit into our general geometric framework. One can use our proposed results -- namely, that the reflections from the tips of the cracks are weaker (in a certain sense) than the reflections from the smooth parts -- combined with the very precise understanding of reflections from the smooth parts, to locate the cracks. The rough conclusion: the tips can be ignored.