Schrodinger operators on periodic discrete graphs

We consider Schrodinger operators with periodic potentials on periodic discrete graphs. It is well known that the spectrum of the Schrodinger operator consists of a finite number of bands, where degenerate bands are eigenvalues of infinite multiplicity. We obtain a localization of each bands in terms of eigenvalues of the Schrodinger operator on a finite graph. We also estimate the Lebesgue measure of the spectrum in terms of geometric invariants for periodic graphs. Moreover, we show that these estimates become identities for specific graphs. The proof is based on Floquet theory and a special gauge transformation of fiber Schrodinger operators.