# Analyticity of the spectrum and Dirichlet-to-Neumann operator technique for quantum graphs

## TL;DR

This paper proves the analyticity of the spectrum of quantum graphs across the entire parameter space and refines the Dirichlet-to-Neumann technique to avoid coordinate-related issues, enabling broader applicability.

## Contribution

It establishes the full analyticity of the dispersion relation on the Grassmannian and improves the Dirichlet-to-Neumann method by removing coordinate-dependent limitations.

## Key findings

- Spectrum analyticity holds over the entire Grassmannian.
- The Dirichlet-to-Neumann technique can be extended beyond coordinate restrictions.
- Addressed issues at the Dirichlet spectrum related to parameter space coordinates.

## Abstract

In some previous works, the analytic structure of the spectrum of a quantum graph operator as a function of the vertex conditions and other parameters of the graph was established. However, a specific local coordinate chart on the Grassmanian of all possible vertex conditions was used, thus creating an erroneous impression that something ``wrong'' can happen at the boundaries of the chart. Here we show that the analyticity of the corresponding ``dispersion relation'' holds over the whole Grassmannian, as well as over other parameter spaces. We also address the Dirichlet-to-Neumann (DtN) technique of relating quantum and discrete graph operators, which allows one to transfer some results from the discrete to the quantum graph case, but which has issues at the Dirichlet spectrum. We conclude that this difficulty, as in the first part of the paper, stems from the use of specific coordinates in a Grassmannian and show how to avoid it to extend some of the consequent results to the general situation.