Zeros of Eigenfunctions of Metric Graphs and Their Applications to Spectral Gap Estimates and to Buckling of Structures

几个世纪以来，板和膜振动模式的节点集（或零点）一直吸引着科学家们——达芬奇、伽利略和胡克的作品中都提到了早期的实验观察。节点集可以揭示底层振动体结构的重要信息。一维本征函数的节点集的数学（对于振动弦）是很容易理解的；然而，在二维或更高的维度中，甚至一些非常简单的问题仍然没有答案。本研究项目的主题是一维弦的网络，也称为度量图。该项目旨在回答有关度量图的振动模态分析的几个相关问题。正在研究的一些问题有许多潜在的应用，从量子物理学（光波导和光学半导体器件网络上孤子解的稳定性）到土木工程（金属框架的屈曲分析），而其他问题则旨在开发对该领域有用的基本工具。度量图在流形光谱分析的广阔领域中占有特殊的地位。一方面，它们易于形象化和数值化研究。另一方面，它们经常显示在更复杂的系统中发现的特征，例如恒定负曲率的流形，并且它们也有大量的应用。该项目将解决三组问题。第一组是一级扰动（如强制零）的最优位置。它有几个潜在的应用，包括理论（了解节点数）和实际（优化谱隙以调整网络结构的稳定性）。第二组问题涉及可以从节点计数中收集到的关于图的信息类型。第三组问题本质上更基本，因为它为其他两组问题创造了工具：首席研究员将根据图上的秩和签名对扰动进行分类，研究图上算子的奇异极限，因为一些边收缩到零，并研究图上的节点计数和散射相位之间的联系。使用的数学技术包括谱分析、微局部分析、图论、辛几何和组合学。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。

The nodal sets (or zeros) of vibrational modes of plates and membranes have been fascinating scientists for centuries -- early experimental observations were mentioned in the works of da Vinci, Galileo, and Hooke. Nodal sets can reveal important information about the structure of the underlying vibrating body. The mathematics of the nodal sets of eigenfunctions in dimension one (for a vibrating string) is well understood; however, in dimensions two and higher even some very simple questions remain unanswered. The subject of this research project is a network of one-dimensional strings, also known as a metric graph. The project aims to answer several interrelated questions concerning vibrational mode analysis on metric graphs. Some questions under study have many potential applications, from quantum physics (stability of soliton solutions on networks of optical waveguides and optical semiconducting devices) to civil engineering (buckling analysis of metal frames), while others are aimed at development of fundamental tools useful to the area. Metric graphs occupy a special niche in the wider field of spectral analysis on manifolds. On one hand, they are easy to visualize and to study numerically. On the other hand, they often display features found in more complicated systems, such as manifolds of constant negative curvature, and they also have a plethora of applications. Three sets of questions will be addressed in the project. The first set is the optimal placement of a rank one perturbation (such as a forced zero). It has several potential applications, both theoretical (understanding the nodal count) and practical (optimizing the spectral gap to tune stability of a network-like structure). The second set of questions addresses the type of information about the graph that can be gleaned from the nodal count. The third set of questions is more fundamental in nature as it creates tools for the other two sets: the principal investigator will classify perturbations on graphs by their rank and signature, study the singular limits of operators on graphs as some edges shrink to zero, and investigate a connection between the nodal count and the scattering phase on a graph. Mathematical techniques to be used include spectral analysis, microlocal analysis, graph theory, symplectic geometry, and combinatorics.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.