On the Spectra of Schrödinger Operators on Zigzag Nanotubes with Multiple Bonds

. In this paper, we study the spectral structure of periodic Schro¨dinger operators on a generalization of carbon nanotubes from the point of view of the quantum graphs. Since there exist chemical double bonds between carbon atoms on a hexagonal lattice with a cylindrical structure corresponding to carbon nanotubes, we study the spectral structure of periodic Schro¨dinger operators on zigzag nanotubes with multiple bonds of atoms in this paper. Utilizing the Floquet–Bloch theory, the spectrum of the operator consists of the absolutely continuous spectral bands and the flat band. We study the relationship between the number of the chemical bonds and the existence of spectral gaps.