Harmonic Analysis of Waves and Eigenfunctions

这个数学研究项目涉及特征函数的行为和传播波在不同的设置，其中经典几何光学的方法不适用。研究的焦点之一是黎曼度规不规则性较低的弯曲空间。对于非二次可微的度量，测地线流不是适定的，因此不能决定波在这样的空间上的能量流。我们考虑李普希茨度量（即一个有界导数），并表明人们仍然可以以与测地线流的不确定性相一致的方式控制能量的色散率。我们应用组合参数得到了特征函数在勒贝格空间中的最佳可能界。本文还研究了具有边界的流形，其中边界外的能量反射由Dirichlet或Neumann条件指定。在这里，衍射效应和多重反射测地线导致能量色散的标准速率的违反。我们采用了为利普希茨度量开发的工具，并将它们与反射技术相结合，以获得最优边界，以及波和薛定谔方程在有边界流形上的新色散估计。这些估计的应用包括非线性波方程和薛定谔方程的新的适定性结果。我们所有工作的一个关键方法是将波分解成适当缩放的波包。我们将这种分解与准微分技术结合起来，在这些情况下获得解的定量界限，在这些情况下，更精确的几何光学渐近公式不成立。振动模式和波传播的研究在物理方程、信号分析和用于地球物理勘探的地震成像方法中都有应用。这个数学研究项目的目的是建立定量界限，这对高频振动的行为既有理论意义，也有实际意义，特别是对地震成像和更好地理解由计算近似引入的误差的性质。我们研究的一种情况是粗糙介质，其特点是声速以非光滑的方式变化，这是地球内部发生的复杂物质混合的模型。我们还研究了波反射坚硬物体的方式，包括涉及复杂多重反射的形状。在这两种情况下，不存在精确的波前分析方法。相反，我们的工作依赖于将信号分解成相干波包，并确定这些包在媒体中传播时的演变（以合适的近似顺序）。在我们的工作中开发的数据包也在医学成像中找到了应用，包括减少感兴趣特征所需的扫描量。

This mathematics research project concerns the behavior of eigenfunctions and propagating waves in various settings where the methods of classical geometric optics do not apply. One focus of study is curved spaces where the Riemannian metric is of low regularity. For metrics which are not twice-differentiable the geodesic flow is not well-posed, and hence does not determine the flow of energy for waves on such spaces. We consider Lipschitz metrics (i.e. one bounded derivative), and show that one can nevertheless control the rate of dispersion of energy in a manner consistent with the uncertainty of the geodesic flow. We apply combinatorial arguments to obtain best possible bounds in Lebesgue spaces for eigenfunctions. Manifolds with boundary, where energy reflection off the boundary is specified by Dirichlet or Neumann conditions, are also studied. Here, diffraction effects and multiply reflected geodesics lead to violation of the standard rate of energy dispersion. We adapt tools developed for Lipschitz metrics, and combine them with reflection techniques to obtain optimal bounds, as well as new dispersive estimates for both wave and Schrodinger equations on manifolds with boundary. Application of these estimates include new well-posedness results for nonlinear wave and Schrodinger equations. A key method for all of our work is the decomposition of waves into appropriately scaled wave packets. We combine this decomposition with paradifferential techniques to obtain quantitative bounds on solutions in these settings, where the more precise asymptotic formulas of geometric optics do not hold.The study of vibrational modes and wave propagation finds applications in the equations of physics, in signal analysis, and in the seismic imaging methods used for geophysical exploration. This mathematics research project aims to establish quantitative bounds that are both of theoretical interest, for the behavior of high-frequency vibrations, and of practical interest, in particular for seismic imaging and for better understanding the nature of the errors that are introduced by computational approximations. One setting we study is that of rough media, characterized by the sound speed changing in a non-smooth manner, which is a model for the intricate mix of materials occurring within the earth. We also study the manner in which waves reflect off hard objects, including shapes which involve complex multiple reflections. In both of these settings, precise wavefront analysis methods do not exist. Instead, our work relies on decomposing signals into coherent wave-packets, and determining (to a suitable order of approximation) the evolution of these packets as they propagate through the media. The packets developed in our work have also found applications in medical imaging, including reducing the amount of scans needed to sense features of interest.