Estimates of bands for Laplacians on periodic equilateral metric graphs

Abstract

We consider Laplacians on periodic equilateral metric graphs. The spectrum of the Laplacian consists of an absolutely continuous part (which is a union of an infinite number of non-degenerated spectral bands) plus an infinite number of flat bands, i.e., eigenvalues of infinite multiplicity. We estimate the Lebesgue measure of the bands on a finite interval in terms of geometric parameters of the graph. The proof is based on spectral properties of discrete Laplacians.