Simplicity of eigenvalues and non-vanishing of eigenfunctions of a quantum graph

Abstract

We prove that after an arbitrarily small adjustment of edge lengths, the spectrum of a compact quantum graph with $\delta$-type vertex conditions can be simple. We also show that the eigenfunctions, with the exception of those living entirely on a looping edge, can be made to be non-vanishing on all vertices of the graph. As an application of the above result, we establish that the secular manifold (also called "determinant manifold") of a large family of graphs has exactly two smooth connected components.

18 pages, 12 figures

References in corpus (2)

• On the Spectral Gap of a Quantum Graph
• Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs

Cited by in corpus (1)

• On the lack of bound states for certain NLS equations on metric graphs