On the spectra of periodic discontinuous quantum graphs

We consider periodic Schrödinger operators on the hexagonal lattice with self-adjoint vertex conditions that allow discontinuity and concentrated mass at the vertices. This model generalizes the periodic Schrödinger operator on the hexagonal lattice with Neumann vertex conditions, known as the graphene Hamiltonian quantum graph. After formulating the corresponding Hamiltonian as a quantum graph and introducing the self adjoint vertex conditions, we provide its dispersion relation, spectrum, eigenvalues, and Dirac points. We also give explicit formulations of our results for the corresponding free operator (zero potential) and show that Borg's theorem is not valid for the Hamiltonians we study, that is, non-degenerate spectral gaps exist for the free operator with our vertex conditions.