Zigzag periodic nanotube in magnetic field

We consider the magnetic Schr\"odinger operator on the so-called zigzag periodic metric graph (a quasi 1D continuous model of zigzag nanotubes) with a periodic potential. The magnetic field (with the amplitude $B\in R$) is uniform and it is parallel to the axis of the nanotube. The spectrum of this operator consists of an absolutely continuous part (spectral bands separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove that all resonances are real. We determine the asymptotics of the periodic and anti-periodic spectrum and of the resonances at high energy. We show that endpoints of the gaps are periodic or anti-periodic eigenvalues or resonances (real branch points of the Lyapunov function). We describe the spectrum as functions of $B$. For example, if $B\to B_{k,m}=\frac{4(\frac{\pi}{2}-\frac{\pi k}{N}+\pi m)}{\sqrt 3 \cos \frac{\pi}{2N}}, k=1,2,..,N, m\in Z$, then some spectral band shrinkes into a flat band i.e., an eigenvalue of infinite multiplicity.