Spectral properties of magnetic chain graphs

TL;DR

This paper investigates the spectral characteristics of a quantum particle on a chain graph of rings under magnetic influence, analyzing effects of various perturbations and confirming a conjecture for periodic cases.

Contribution

It provides a detailed spectral analysis of magnetic chain graphs with delta couplings, including perturbation effects and a proof of the Saxon-Hutner conjecture for periodic perturbations.

Findings

Spectral properties are characterized for unperturbed and perturbed systems.

Weak local and periodic perturbations significantly influence the spectrum.

The Saxon-Hutner conjecture is proven for periodic perturbations.

Abstract

We discuss spectral properties of a charged quantum particle confined to a chain graph consisting of an infinite array of rings under influence of a magnetic field assuming a $\delta$-coupling at the points where the rings touch. We start with the situation when the system has a translational symmetry and analyze spectral consequences of perturbations of various kind, such as a local change of the magnetic field, of the coupling constant, or of a ring circumference. A particular attention is paid to weak perturbations, both local and periodic; for the latter we prove a version of Saxon-Hutner conjecture.