Harmonic Analysis of Waves and Eigenfunctions

这个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。首席研究人员正在引入新的数学工具，将对能量流动现象的理解扩展到标准分析工具无法处理的区域。重点放在使用适合计算分析的方法上。谐波分析工具在信号和图像处理中的应用已有很长的历史，研究人员的工作包括开发类似的工具来研究波的传播。该项目的一个重点是凸起物体对波的反射。这样做的目的是为了更准确地了解波在与边界相互作用时如何扩散。应用包括控制振动模式在凸形物体边界附近的集中程度。该项目的另一个重点是研究地震波，更广泛地说，是研究弹性介质中的波。地震波可以同时包括横向和纵向位移，并且这些波的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的模式在通过高度不均匀的介质传播时相互作用的顺序，例如地球内发生的物质混合物。实际影响包括对单独处理模式的计算模型的误差估计，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。新的方法也将被用于研究衰变的振动模式，即所谓的共振态。具有共振态的系统的例子包括微波腔和具有势垒的量子力学系统。谐和分析的工具被用来研究共振的存在，并将共振次数与系统的性质联系起来。这个项目的所有结果都将通过开放获取的网站在网上发布。这个项目涉及到使用调和分析技术来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，在各种几何光学的传统数学方法不适用的情况下，波的色散速率仍然与几何光学预测的相同。所研究的背景的一个例子是粗糙媒体，由具有两次可微的度量的流形建模。通过这种介质的波的能量可能会散射，只有关于能量流动的不精确知识可用。然而，主要研究人员的工作表明，人们可以对这种介质中的能量流进行充分的控制，以建立重要的结果，例如对非线性波动和薛定谔方程领域感兴趣的色散估计。一个相关的应用是限制这种介质中的本征函数可以集中的程度。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。这位首席研究员的研究调查了各种地震模式之间的能量转移，这些模式是由传播它们的介质中的奇点引起的。凸形障碍物对波的散射是该项目的另一个重点。在拟议研究的这一部分中，目标是获得边界小区域的精确能量衰减率。结果表明，能量不可能在边界附近集中到比几何光学预测的程度更高的程度，这将导致新的色散估计，从而为含障碍区域上的非线性方程的研究带来结果。研究人员还将调和分析应用到有界势薛定谔算子的共振研究中。建立了位势的高阶Soblev正则性与正则化热迹的级数展开式之间的密切联系，并用它证明了这类位势的共振的存在性。

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.