论文标题
公制图上的本地特征向量
Localized eigenvectors on metric graphs
论文作者
论文摘要
使用我们先前发表的算法，我们分析了在实际应用中发生的两个度量图，分析了广义拉普拉斯的特征向量。如预期的那样，特征向量的定位很少，并且应该调整网络以观察到局部特征向量。我们得出共振条件，以获取各种几何构型及其组合以形成更复杂的谐振结构的局部特征向量。这些本地化特征向量提出了基于$ L_2 $ NORM的新本地化指标。即使在泄漏的边界条件下，它们也可以激发，如公制图上的时间依赖性波方程的数值解所示。最后，该研究提出了基于度量图制作共鸣系统的实用方法。
Using our previously published algorithm, we analyze the eigenvectors of the generalized Laplacian for two metric graphs occurring in practical applications. As expected, localization of an eigenvector is rare and the network should be tuned to observe exactly localized eigenvectors. We derive the resonance conditions to obtain localized eigenvectors for various geometric configurations and their combinations to form more complicated resonant structures. These localized eigenvectors suggest a new localization indicator based on the $L_2$ norm. They also can be excited, even with leaky boundary conditions, as shown by the numerical solution of the time-dependent wave equation on the metric graph. Finally, the study suggests practical ways to make resonating systems based on metric graphs.
