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。波动方程和薛定谔方程是描述波（包括声波、光波和引力波）如何在自然界中传播的重要偏微分方程。 当底层介质是光滑的时，关于这些方程的解的行为已经知道得很多。然而，大多数实际应用涉及在包含障碍物或不连续性的介质中传播的波。该项目的目的是解决关于当底层介质不光滑时波如何散射和衰减的公开问题。该项目的结果预计将为涉及非光滑介质中波动行为的几个应用的预测提供信息，包括地震波的传播，等离子体的行为以及光纤中的光传播。该项目将有助于在数学物理的一个中心，活跃的领域的本科生和研究生的培训。该项目有两个目标。第一个是证明半经典薛定谔算子在有限正则性下的预解式的上界。这样的预解边界是粗糙波方程局部能量衰减率的前兆。第二个目标是使用最近发展的Fredholm方法建立散射型本征函数的存在性的广泛的一类非线性Helmholtz方程。本研究的主要工具来自于半经典和微局域分析。特别是，首席研究员的目标是扩展积极的交换子和Carleman估计，分离的变量，和b-向量场分析薛定谔算子与低正则性系数。这将产生高频率的预解估计，并反过来精确的能量衰减率波在异质介质中传播。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.