  We consider the Schr\"odinger operator with a periodic potential on quasi-1D
models of armchair single-wall nanotubes. The spectrum of this operator
consists of an absolutely continuous part (intervals separated by gaps) plus an
infinite number of eigenvalues with infinite multiplicity. We describe the
absolutely continuous spectrum of the Schr\"odinger operator: 1) the
multiplicity, 2) endpoints of the gaps, they are given by periodic or
antiperiodic eigenvalues or resonances (branch points of the Lyapunov
function), 3) resonance gaps, where the Lyapunov function is non-real. We
determine the asymptotics of the gaps at high energy.
