Always detectable eigenfunctions on metric graphs

TL;DR

This paper proves that Laplacians on metric graphs have an infinite sequence of eigenvalues with eigenfunctions that are non-zero at all non-Dirichlet vertices, extending previous results on metric trees.

Contribution

It establishes the existence of always detectable eigenfunctions on general metric graphs, generalizing prior work on metric trees.

Findings

Infinite sequence of simple eigenvalues proven

Eigenfunctions are non-zero at all non-Dirichlet vertices

Extends previous results from metric trees to arbitrary metric graphs

Abstract

It is proven following [18[ that Laplacians with standard vertex continuous on metric trees and with standard and Dirichlet conditions on arbitrary metric graphs possess an infinite sequence of simple eigenvalues with the eigenfunctions not equal to zero in any non-Dirichlet vertex.