Collaborative Research: Microlocal Concentration and Propagation in Spectral Theory

在自然界中，波型方程的解出现在量子粒子的波函数、音乐厅的声学和引力波的散射等现象中。在这些情况下，有一定的基本频率和振动模式与空间的基本几何形状相关。这些频率和模态在理解波动现象的行为方面起着至关重要的作用。例如，它们出现在音乐厅的声学设计和乐器的振动中。该项目开发了基于传播现象的工具，以研究空间几何形状与相应模式行为之间的微妙关系。该研究将为一般空间的振动模态结构提供新的认识。对于非正曲率空间，许多定量规律性结果（由L^p范数测量）已经为人所知。然而，在限制较少的几何假设下，这些类型的估计是不可用的。这个项目的主要目标有两个方面：(1)描述导致单个模态L^p范数增长的集中现象，(2)利用这种理解，仅基于底层流形的动态特性，对模态和频率进行估计。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。

In nature, solutions of wave-type equations appear in phenomena as far reaching as wave functions of quantum particles, the acoustics of concert halls, and scattering of gravitational waves. In these situations, there are certain fundamental frequencies and vibrational modes associated to the underlying geometry of the space. These frequencies and modes play an essential role in the understanding of the behavior of wave phenomena. They appear, for example, in the acoustical design of concert halls and the vibration of instruments. This project develops tools, based on propagation phenomena, to study the delicate relationship between the geometry of the space and behavior of the corresponding modes. This research will provide new insight into the structure of vibrational modes for general spaces. Many quantitative regularity results (as measured by L^p norms) are already known for spaces with non-positive curvature. However, these types of estimates are unavailable under less restrictive geometric assumptions. The main goals of this project are two fold: (1) to describe the concentration phenomena that result in L^p norm growth for individual modes, and (2) to use this understanding to produce estimates for modes and frequencies based solely on dynamical properties of the underlying manifold.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.