Harmonic Analysis of Waves and Eigenfunctions

该项目研究物理系统的各种数学模型的能量流动和振动模式的行为。首席研究员正在引入新的数学工具，将对能量流现象的理解扩展到标准分析工具无法处理的领域。重点放在使用的方法，借给自己的计算分析。谐波分析工具在信号和图像处理中的应用历史悠久，研究人员的工作涉及开发用于研究波传播的类似工具。该项目的重点之一是凸形物体的波反射。目的是更精确地了解波在与边界相互作用时如何分散。应用包括控制振动模式可以集中在凸形物体边界附近的程度。该项目的另一个重点是研究地震波，以及更一般的弹性介质中的波。地震波可以涉及横向和纵向位移，并且波的这些分量通常以不同的速度传播。该项目的一个目标是估计这些不同模式在通过高度异质介质（例如地球内发生的材料混合物）传播时相互作用的顺序。实际影响包括估计的计算模型，分别对待模式的错误，以及是否有必要包括它们的相互作用，以达到指定的准确度。新的方法也将用于研究衰减的振动模式，称为共振态。具有谐振态的系统的示例包括微波腔和具有势垒的量子力学系统。谐波分析的工具被用来研究共振的存在，并将共振的数量与系统的性质联系起来。该项目的所有成果将通过开放式网站在线传播。该项目涉及使用谐波分析技术，以促进我们对非均匀介质中的波和本征函数的理解。该项目的一个主要目标是表明，在几何光学的传统数学方法不适用的各种设置中，波的色散率仍然与几何光学预测的相同。一个例子的设置研究是粗糙的介质，模拟流形与二次微分度量。通过这种介质的波的能量可以散射，并且只有关于能量流的不精确的知识可用。尽管如此，主要研究者的工作表明，人们可以获得足够的控制能量流在这种介质中建立重要的结果，如色散估计是感兴趣的领域的非线性波和薛定谔方程。一个相关的应用是限制这种介质中本征函数的集中程度。另一个非常重要的例子是地震波，它可以以不同的速度传播，这取决于初始位移的性质。首席研究员的研究调查了各种地震模式之间的能量转移，这些模式是由它们传播的介质中的奇异性引起的。波浪从凸形障碍物上的散射是该项目的另一个重点。在这一部分的拟议研究的目标是获得精确的能量衰减率在小区域的边界。结果将表明，能量不能集中在边界附近的程度高于几何光学预测，并会导致新的色散估计，随之而来的结果的非线性方程的研究领域的障碍。研究人员还将调和分析应用于有界势的薛定谔算子的共振研究。高Sobolev正则性的潜力，和系列的正规化热迹的展开，建立了一个尖锐的关系，然后被用来证明这种潜力的共振的存在。

This project studies the flow of energy, and the behavior of vibrational modes, for various mathematical models of physical systems. The principal investigator is introducing new mathematical tools that extend the understanding of energy flow phenomena to regimes that cannot be handled by standard analytical tools. Emphasis is placed on using methods that lend themselves to computational analysis. There is a long history of harmonic analysis tools finding application in signal and image processing, and the investigator's work involves developing similar tools for the study of wave propagation. One focus of the project is the reflection of waves from convex objects. The goal is to obtain a more precise understanding of how the wave disperses as it interacts with the boundary. Applications include control on the degree to which vibrational modes can concentrate near the boundary of a convex object. The project has another focus in the study of seismic waves, and more generally waves in elastic media. Seismic waves can involve both transverse and longitudinal displacements, and these components of the wave generally propagate at different speeds. A goal of the project is to estimate the order to which these distinct modes interact with each other as they propagate through highly heterogeneous media, such as the mixture of materials occurring within the earth. Practical implications include estimates on the error for computational models that treat the modes separately, and whether it is necessary to include their interaction in order to attain an assigned degree of accuracy. New methods will also be used in the study of decaying vibrational modes, known as resonant states. Example of systems with resonant states include microwave cavities, and quantum mechanical systems with potential barriers. Tools from harmonic analysis are used to study the existence of resonances, and to relate the number of resonances to properties of the system. All results of this project will be disseminated online, through open access websites.This project involves the use of harmonic analysis techniques to advance our understanding of waves and eigenfunctions in nonhomogeneous media. A main goal of the project is to show that, in various settings where the traditional mathematical methods of geometric optics do not apply, the rate of dispersion of waves is nevertheless the same as would be predicted by geometric optics. An example of a setting studied is rough media, modeled by manifolds with twice-differentiable metrics. The energy of waves passing through such media can scatter, and only imprecise knowledge on energy flow is available. Nevertheless, the principal investigator's work shows that one can obtain sufficient control on energy flow in such media to establish important results, such as dispersive estimates that are of interest in the fields of nonlinear wave and Schrodinger equations. A related application is bounding the degree to which eigenfunctions in such media can concentrate. Another example of significant importance is seismic waves, which can propagate at distinct speeds, depending on the nature of the initial displacement. The principal investigator's research investigates the transfer of energy between various seismic modes that is induced by the singularities in the media through which they propagate. Scattering of waves from convex obstacles is another focus of the project. In this part of the proposed research the goal is to obtain precise rates of energy decay in small regions of the boundary. The results would show that energy cannot concentrate near the boundary to a degree higher than predicted by geometric optics, and would lead to new dispersive estimates, with consequent results for the study of nonlinear equations on domains with obstacles. The investigator is also applying harmonic analysis to the study of resonances for Schrodinger operators with bounded potentials. A sharp relation between higher Sobolev regularity of the potential, and series expansions for the regularized heat trace, is established, which is then used to prove the existence of resonances for such potentials.