Spectra of semi-infinite quantum graph tubes

TL;DR

This paper analyzes the spectral properties of semi-infinite quantum graph tubes formed by rolling up doubly periodic graphs, revealing how eigenfunctions relate to Floquet modes and how the spectrum is influenced by boundary conditions.

Contribution

It provides a detailed analysis of the spectrum and eigenfunctions of semi-infinite quantum graph tubes, including the complex dispersion relation and Floquet mode structure.

Findings

Spectrum includes eigenvalues depending on boundary conditions.

Eigenfunctions involve all Floquet modes of the full tube.

The number of Floquet modes for a given eigenvalue is twice the maximum of lpha and eta.

Abstract

The spectrum of a semi-infinite quantum graph tube with square period cells is analyzed. The structure is obtained by rolling up a doubly periodic quantum graph into a tube along a period vector and then retaining only a semi-infinite half of the tube. The eigenfunctions associated to the spectrum of the half-tube involve all Floquet modes of the full tube. This requires solving the complex dispersion relation $D(\lambda,k_1,k_2)=0$ with $(k_1,k_2)\in(\mathbb{C}/2\pi\mathbb{Z})^2$ subject to the constraint $\alpha k_1 + \beta k_2 \equiv 0$ (mod $2\pi$), where $\alpha$ and $\beta$ are integers. The number of Floquet modes for a given $\lambda\in\mathbb{R}$ is $2\max\left\{ \alpha, \beta \right\}$. Rightward and leftward modes are determined according to an indefinite energy flux form. The spectrum may contain eigenvalues that depend on the boundary conditions, and some eigenvalues may be embedded in the continuous spectrum.