Abstract
We consider Laplacians on periodic discrete graphs. The spectrum of the Laplacian consists of a finite number of bands, where degenerate bands are eigenvalues of infinite multiplicity. We introduce a new invariant I for periodic graphs and obtain a decomposition of the Laplacian into a direct integral, where fiber Laplacians (matrices) have the minimal number (≤ 2I) of coefficients depending on the quasimomentum. Using this decomposition, we estimate the position of each band and the Lebesgue measure of the Laplacian spectrum in terms of the new invariants. Moreover, similar results for Schrodinger operators with periodic potentials are obtained.
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