On the spectrum of a bent chain graph

TL;DR

This paper investigates how bending a quantum chain graph with delta couplings introduces eigenvalues in spectral gaps, analyzing their dependence on system parameters and the effects of bending on spectral properties.

Contribution

It introduces a model of a bent quantum chain graph with delta couplings and analyzes how bending induces eigenvalues in spectral gaps, extending understanding of spectral effects of geometric deformations.

Findings

Bending creates eigenvalues in spectral gaps.

Eigenvalues depend on coupling strength and bending angle.

Resonances are influenced by the bending deformation.

Abstract

We study Schr\"odinger operators on an infinite quantum graph of a chain form which consists of identical rings connected at the touching points by $\delta$-couplings with a parameter $\alpha\in\R$. If the graph is "straight", i.e. periodic with respect to ring shifts, its Hamiltonian has a band spectrum with all the gaps open whenever $\alpha\ne 0$. We consider a "bending" deformation of the chain consisting of changing one position at a single ring and show that it gives rise to eigenvalues in the open spectral gaps. We analyze dependence of these eigenvalues on the coupling $\alpha$ and the "bending angle" as well as resonances of the system coming from the bending. We also discuss the behaviour of the eigenvalues and resonances at the edges of the spectral bands.