High energy spectral and scattering phenomena via microlocal analysis

亥姆霍兹方程的解在光谱理论中起着至关重要的作用。在自然界中，这些函数出现在各种现象中，包括量子粒子的波函数、热传导、振动膜的剖面、音乐厅的声学和引力波的传播。因此，了解它们的行为在数学物理学中具有基本的重要性，至少从17世纪末Chladni的工作开始就开始研究了。对Helmholtz方程高能解的浓度性质的研究，此后被称为本征函数或振动模，是非常不平凡的，一直是数学界广泛工作的主题。这个项目延续了这一悠久的传统，并将推动我们目前对振动模式的理解的界限。该项目将回答这样的问题：一种模式随着能源的增长能以多快的速度增长？这种模式能在多大程度上集中精力？这种高能模式的典型行为是什么？当模式非常集中时，它们看起来是什么样子的？在环境的微小扰动下，振动模式的快速增长能持续吗？除了这些关于物理上受限的振动模式的问题外，本征函数(或其推广)可以用来理解当能量可以逃逸到无穷大时波的长期行为。令人惊讶的是，即使是两个玻璃球的光散射的数学也没有得到正确的理解。该项目旨在开发研究散射波的新工具，以解决这一问题。此外，该项目旨在了解具有准周期结构的材料的性质。这些结构在自然界中无处不在，但人们对它们的性质仍然知之甚少。

Solutions to the Helmholtz equation play a crucial role in spectral theory. In nature, these functions appear in phenomena as far reaching as the wave functions of quantum particles, heat conduction, profiles of vibrating membranes, the acoustics of concert halls and the propagation of gravitational waves. Understanding their behaviour is therefore of fundamental importance in mathematical physics and has been studied since at least the work of Chladni in the late 1700s. The study of concentration properties of high energy solutions to the Helmholtz equation, henceforth called eigenfunctions or vibrational modes, is highly non-trivial and has been the subject of extensive work in the mathematics community. This project continues this long tradition and will push the boundaries of our current understanding of vibrational modes. The project will answer questions like: How fast can a mode grow with energy? How physically concentrated can this mode be? What is the typical behavior of such a high energy mode? When modes are extremely concentrated, what do they look like? Can rapid growth of vibrational modes persist under small perturbations of the environment?In addition to these questions about vibrational modes which are physically confined, eigenfunctions (or generalizations there-of) can be used to understand the long time behavior of waves when energy can escape to infinity. Surprisingly, even the mathematics of light scattering off of two glass spheres is not properly understood. This project aims to develop new tools for the study of scattered waves that will address this problem. Moreover the project aims to understand properties of materials with quasiperiodic structure. These structures are ubiquitous in nature, but, nevertheless, their properties remain poorly understood.