Surgery of Graphs: M-Function and Spectral Gap

Abstract

We discuss behaviour of the spectral gap for quantum graphs when two metric graphs are glued together. It appears that precise answer to this question can be given using a natural generalisation of the Titchmarsh-Weyl M -functions.

On isospectral metric graphs

“…The following two formulas express the M -functions through the traces of the standard (satisfying standard vertex conditions on ∂Γ and Dirichlet (satisfying Dirichlet conditions on ∂Γ) [18,19,22] (5)…”

On Magnetic Boundary Control for Metric Graphs

“…The structure of M -functions for the graph is best seen from the following two explicit formulas [30,31]…”

Always detectable eigenfunctions on metric graphs

“…One such property is that the eigenfunctions may have support not coinciding with the whole graph, or may just vanish at the vertices leading to problems when defining nodal domains. Moreover, if one of the eigenfunctions is vanishing at a vertex V 0 , then it is not seen in the Titchmarsh-Weyl M -function associated with this vertex [9,12]:…”