Spectra of magnetic chain graphs: coupling constant perturbations

TL;DR

This paper investigates the spectral characteristics of a magnetic quantum ring chain graph, focusing on how vertex coupling perturbations influence the band and discrete spectra, with detailed analysis of specific perturbation scenarios.

Contribution

It provides a detailed analysis of spectral changes in a magnetic quantum graph due to finite vertex coupling perturbations, including asymptotic and distant perturbation effects.

Findings

Band spectrum under translational symmetry identified

Discrete spectrum in gaps affected by vertex perturbations

Asymptotic behavior analyzed for weak and distant perturbations

Abstract

We analyze spectral properties of a quantum graph in the form of a ring chain with a $\delta$ coupling in the vertices exposed to a homogeneous magnetic field perpendicular to the graph plane. We find the band spectrum in the case when the chain exhibits a translational symmetry and study the discrete spectrum in the gaps resulting from changing a finite number of vertex coupling constants. In particular, we discuss in details some examples such as perturbations of one or two vertices, weak perturbation asymptotics, and a pair of distant perturbations.