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

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 were established. However, a specific local coordinate chart on the Grassmannian 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 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.