Magnetic Schr\"odinger operators on armchair nanotubes

We consider the Schr\"odinger operator with a periodic potential on a quasi 1D continuous periodic model of armchair nanotubes in $\R^3$ in a uniform magnetic field (with amplitude $B\in \R$), which 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 eigenfunctions with the same eigenvalue including compactly supported. We describe the spectrum as a function of $B$. For some specific potentials we prove an existence of gaps independent on the magnetic field. If $B\ne 0$, then there exists an infinite number of gaps $G_n$ with the length $|G_n|\to\iy$ as $n\to\iy$, and we determine the asymptotics of the gaps at high energy for fixed $B$. Moreover, we determine the asymptotics of the gaps $G_n$ as $B\to 0$ for fixed $n$.