Harmonic Analysis of Waves and Eigenfunctions

波和本征函数的谐波分析

• 批准号: 1500098
• 负责人: Hart Smith
• 金额: $ 29.58万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500098&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Waves; Eigenfunctions

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.

这个项目研究能量的流动，和振动模式的行为，为物理系统的各种数学模型。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，研究者的工作包括开发用于波传播研究的类似工具。该项目的一个重点是来自凸物体的波的反射。目标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，以及更普遍的弹性介质中的波。地震波可以包括横向和纵向位移，这些波的分量通常以不同的速度传播。该项目的目标是估计这些不同模式在高度异质介质中传播时相互作用的顺序，例如在地球内部发生的材料混合物。实际意义包括对单独处理模态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精度程度。新的方法也将用于研究衰减的振动模式，即共振态。共振态系统的例子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性，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获取网站在线发布。这个项目涉及使用谐波分析技术来推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情况下，波的色散速率仍然与几何光学预测的速率相同。所研究的一个例子是粗糙介质，由具有两次可微度量的流形建模。通过这种介质的波的能量会分散，关于能量流的知识只能是不精确的。然而，首席研究员的工作表明，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，例如在非线性波和薛定谔方程领域中感兴趣的色散估计。一个相关的应用是限定特征函数在这种介质中可以集中的程度。另一个重要的例子是地震波，它可以根据初始位移的性质以不同的速度传播。首席研究员的研究调查了各种地震模式之间的能量传递，这是由它们传播的介质中的奇点引起的。来自凸障碍物的波散射是该项目的另一个重点。在这部分研究中，我们的目标是在边界的小区域内获得精确的能量衰减率。结果表明，能量不能在边界附近集中到比几何光学预测的更高程度，并将导致新的色散估计，从而对有障碍区域上的非线性方程的研究产生影响。研究者还将谐波分析应用于有界势薛定谔算符的共振研究。建立了势的高索博列夫规则性与正则化热迹的级数展开之间的明显关系，然后用它来证明这些势的共振的存在。