Geometric Scattering Theory, Resolvent Estimates, and Wave Asymptotics

本研究项目研究波在非光滑介质中的传播。波和薛定谔方程是描述波(包括声波、光和引力波)如何在自然界中传播的重要偏微分方程式。当底层介质是光滑的时，关于这些方程的解的行为已知很多。然而，大多数实际应用涉及到波在含有障碍物或不连续的介质中传播。这个项目的目的是解决当底层介质不光滑时，波如何散射和衰减的公开问题。该项目的结果有望为涉及非光滑介质中的波行为的几个应用领域的预测提供参考，包括地震波的传播、等离子体的行为和光纤中的光传播。该项目将有助于培养本科生和研究生在一个中心的，活跃的数学物理领域。这个项目涉及几何散射理论的研究。该项目有两个目标。首先证明了半经典薛定谔算子在有限正则性下预解的上界。这样的预解边界是粗波方程局部能量衰减率的先兆。第二个目标是利用最近发展起来的Fredholm法来证明一大类非线性Helmholtz方程的散射型本征函数的存在性。这个项目的主要工具来自半经典和微局部分析。特别是，主要研究人员的目标是将正交换子和Carleman估计、分离变量和b向量场分析推广到具有低正则系数的薛定谔算子。这将产生高频分辨率估计，进而为波在不同介质中传播的精确能量衰减率。该奖项反映了NSF的法定使命，通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This research project studies the propagation of waves in non-smooth media. Wave and Schrödinger equations are important partial differential equations that describe how waves, including sound, light, and gravitational waves, propagate in the natural world. Much is known about the behavior of solutions to these equations when the underlying media are smooth. However, most practical applications involve waves propagating in media that contain obstructions or discontinuities. The aim of this project is to resolve open questions about how waves scatter and decay when the underlying medium is not smooth. The results of the project are expected to inform predictions in several applications involving wave behavior in non-smooth media, including the propagation of seismic waves, the behavior of plasma, and light propagation in optical fibers. The project will contribute to the training of undergraduate and graduate students in a central, active area of mathematical physics.This project concerns research in geometric scattering theory. The project has two objectives. The first is to prove upper bounds on the resolvent of the semiclassical Schrödinger operator in limited regularity. Such resolvent bounds are a precursor to local energy decay rates for rough wave equations. The second objective is to use the recently developed Fredholm method to establish the existence of scattering-type eigenfunctions for a wide class of nonlinear Helmholtz equations. The main tools for this project come from semiclassical and microlocal analysis. In particular, the principal investigator aims to extend positive commutator and Carleman estimates, separation of variables, and b-vector field analysis to Schrödinger operators with low regularity coefficients. This will yield high frequency resolvent estimates, and in turn precise energy decay rates for waves traveling in heterogeneous media.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.