Schrödinger operators on zigzag periodic graphs

We consider the Schrödinger operator on the so-called zigzag periodic metric graph (a continuous version of zigzag nanotubes) with a periodic potential. 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 all compactly supported (localization) 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 there exist two types of gaps: 1) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, 2) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We obtain the following results from the inverse spectral theory: 1) we describe all finite gap potentials, 2) the mapping: potential – all eigenvalues is a real analytic isomorphism for some class of potentials.