度量图的局部特征向量
Localized eigenvectors on metric graphs
摘要：
使用我们以前发表的算法，我们分析了在实际应用中出现的两个度量图的广义拉普拉斯算子的特征向量。正如预期的那样，特征向量的局部化是罕见的，并且应该调整网络以观察精确局部化的特征向量。我们推导出共振条件，以获得各种几何配置及其组合，以形成更复杂的共振结构的本地化本征向量。这些本地化特征向量提出了基于$L_2$规范的新本地化指示符。它们也可以被激发，即使有漏边界条件，如度规图上的含时波动方程的数值解所示。最后，研究提出了实用的方法，使谐振系统的基础上度量图。
英文摘要：
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.
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