Linear Partial Differential Equations on Singular Spaces

许多看似不同的物理现象，如光、声音传播和量子粒子的运动，都以数学上统一的方式被描述为传播波。因此，描述波如何振荡和消亡取决于产生它们的源和它们在其中传播的介质，这是相当重要的。该项目的目标是研究围绕波衰减率的几个问题，特别是当波遇到介质中的尖锐不连续性时，它如何受到衍射效应的影响。感兴趣的问题包括那些由非常小尺度物理学（氢原子）和非常大尺度物理学（宇宙学中产生的背景波的衰变）激发的问题。PI先前的工作将继续分析在实际设置中模拟这些方程的解所使用的计算方法的有效性。研究生和博士后研究员的培训将纳入整个项目。这个项目围绕着涉及在几种设置中波的源附近的衰减率的问题。特别感兴趣的将是理解波衍射粗糙介质中的定性行为和长期衰减率的解决方案，波和薛定谔方程的作用。在研究库仑势的奇异性对氢原子狄拉克传播子结构的影响时，衍射效应将再次发挥重要作用。在广义相对论感兴趣的时空中，将研究时空的大尺度结构对波的衰变及其辐射模式的结构的影响。该项目将进一步研究用于计算波散射的数值算法的性能，将以前未在这些问题中使用的相空间分析技术引入其中。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many seemingly different physical phenomena such as light, sound propagation, and the motion of quantum particles are described in a mathematically unified manner as propagating waves. It is therefore of considerable practical importance to describe how waves oscillate and die off depending on the source producing them and the medium in which they propagate. The goal of this project is to investigate several questions surrounding the rate of wave decay, and in particular how it is influenced by the effects of diffraction, which occurs when the waves encounter sharp discontinuities in the medium. The questions of interest include those motivated by very small-scale physics (the hydrogen atom) and very large-scale physics (decay of waves on backgrounds arising in cosmology). The PI’s prior work on analyzing the effectiveness of computational methods used in modeling solutions to these equations in practical settings will be continued. Training of graduate students and postdoctoral fellows will be incorporated throughout the project.This project revolves around questions involving the decay rate of waves near their source in several settings. Of particular interest will be understanding the role of wave diffraction by rough media in the qualitative behavior and long-time decay rates of solutions to wave and Schrödinger equations. When studying the effects of the singularity of the Coulomb potential on the structure of the Dirac propagator for the hydrogen atom, diffractive effects will again play an important role. In spacetimes of interest in general relativity, the large-scale structure of spacetime effects on the decay of waves and the structure of their radiation patterns will be investigated. This project will further entail a study of the performance of numerical algorithms for computation of the scattering of waves, bringing to bear techniques of phase space analysis that have not previously been employed in these problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.