Geometric Scattering Theory, Resolvent Estimates, and Wave Asymptotics

本课题研究的是波在非光滑介质中的传播。波动方程和Schrödinger方程是重要的偏微分方程，它们描述了波（包括声音、光和引力波）在自然界中的传播方式。当底层介质是光滑时，这些方程的解的行为是已知的。然而，大多数实际应用涉及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关于当底层介质不光滑时波是如何散射和衰减的开放性问题。该项目的结果预计将为涉及非光滑介质中的波行为的几个应用提供预测，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数学物理中心活跃领域的本科生和研究生。这个项目涉及几何散射理论的研究。该项目有两个目标。首先证明了有限正则半经典Schrödinger算子解的上界。这种可解边界是粗糙波动方程的局部能量衰减率的前兆。第二个目标是使用最近发展的Fredholm方法来建立一类广泛的非线性亥姆霍兹方程的散射型特征函数的存在性。本项目的主要工具来自半经典和微局部分析。特别地，主要研究者的目标是将正换向子和Carleman估计、变量分离和b向量场分析扩展到具有低正则系数的Schrödinger算子。这将产生高频分解估计，进而精确的能量衰减率波在非均质介质中传播。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.