Spectral gaps and discrete magnetic Laplacians

摘要

The aim of this article is to give a simple geometric condition that guarantees the existence of spectral gaps of the discrete Laplacian on periodic graphs. For proving this, we analyse the discrete magnetic Laplacian (DML) on the finite quotient and interpret the vector potential as a Floquet parameter. We develop a procedure of virtualising edges and vertices that produces matrices whose eigenvalues (written in ascending order and counting multiplicities) specify the bracketing intervals where the spectrum of the Laplacian is localised. We prove Higuchi Shirai's conjecture for Z-periodic trees and apply our technique in several examples like the polypropylene or the polyacetylene to show the existence of spectral gaps. (C) 2018 Elsevier Inc. All rights reserved.