Spectral band localization for Schr\"odinger operators on periodic graphs

Abstract

We consider Schr\"odinger operators on periodic discrete graphs. It is known that the spectrum of these operators has band structure. We obtain a localization of spectral bands in terms of eigenvalues of Dirichlet and Neumann operators on a finite graph, which is constructed from the fundamental cell of the periodic graph. The proof is based on the Floquet decomposition of Schr\"odinger operators and the minimax principle.