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The spectrum of a semi-infinite quantum graph tube with square period cellsis analyzed. The structure is obtained by rolling up a doubly periodic quantumgraph into a tube along a period vector and then retaining only a semi-infinitehalf of the tube. The eigenfunctions associated to the spectrum of thehalf-tube involve all Floquet modes of the full tube. This requires solving thecomplex 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 $\alphak_1 + \beta k_2 \equiv 0$ (mod $2\pi$), where $\alpha$ and $\beta$ areintegers. The number of Floquet modes for a given $\lambda\in\mathbb{R}$ is$2\max\left\{ \alpha, \beta \right\}$. Rightward and leftward modes aredetermined according to an indefinite energy flux form. The spectrum maycontain eigenvalues that depend on the boundary conditions, and someeigenvalues may be embedded in the continuous spectrum.

Spectra of semi-infinite quantum graph tubes