Localized eigenvectors on metric graphs

TL;DR

This paper investigates localized eigenvectors in metric graphs, deriving resonance conditions, proposing a new localization indicator, and demonstrating their excitation and practical applications in resonating systems.

Contribution

It introduces a method to analyze and induce localized eigenvectors in metric graphs, including resonance conditions and a new localization indicator.

Findings

Localized eigenvectors are rare without tuning.

Resonance conditions enable localization in specific configurations.

Localized eigenvectors can be excited even with leaky boundaries.

Abstract

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.