Spectra of semi-infinite quantum graph tubes

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.