On fully supported eigenfunctions of quantum graphs

TL;DR

This paper proves that all tree-shaped metric graphs have a full support orthonormal eigenfunction sequence, revealing that their nodal domain behavior mirrors one-dimensional cases despite complex topologies.

Contribution

It establishes the existence of fully supported eigenfunctions on trees and general graphs with Dirichlet vertices, extending understanding of spectral properties on quantum graphs.

Findings

Eigenfunctions on trees have full support and form an orthonormal sequence.

The nodal domain ratio $ u_n/n$ approaches 1 along a subsequence.

Results extend to graphs with Dirichlet vertices, delta conditions, and constant potentials.

Abstract

We prove that every metric graph which is a tree has an orthonormal sequence of Laplace-eigenfunctions of full support. This implies that the number of nodal domains $\nu_n$ of the $n$-th eigenfunction of the Laplacian with standard conditions satisfies $\nu_n/n \to 1$ along a subsequence and has previously only been known in special cases such as mutually rationally dependent or rationally independent side lengths. It shows in particular that the Pleijel nodal domain asymptotics from two- or higher dimensional domains cannot occur on these graphs: Despite their more complicated topology, they still behave as in the one-dimensional case. We prove an analogous result on general metric graphs under the condition that they have at least one Dirichlet vertex. Furthermore, we generalize our results to Delta vertex conditions and to edgewise constant potentials. The main technical contribution is a new expression for a secular function in which modifications to the graph, to vertex conditions, and to the potential are particularly easy to understand.