Effective masses for Laplacians on periodic graphs

TL;DR

This paper analyzes the spectral properties of Laplacians on periodic graphs, providing estimates for effective masses at spectral band edges using geometric parameters, with implications for understanding wave propagation.

Contribution

It introduces new estimates for effective masses on periodic graphs and relates discrete and metric Laplacians through geometric and spectral analysis.

Findings

Effective mass estimates at spectral band edges.

Two-sided bounds for the beginning of the spectrum.

Relation between discrete and metric graph Laplacians.

Abstract

We consider Laplacians on periodic both discrete and metric equilateral graphs. Their spectrum consists of an absolutely continuous part (which is a union of non-degenerate spectral bands) and flat bands, i.e., eigenvalues of infinite multiplicity. We estimate effective masses associated with the ends of each spectral band in terms of geometric parameters of graphs. Moreover, in the case of the beginning of the spectrum we determine two-sided estimates of the effective mass in terms of geometric parameters of graphs. The proof is based on the Floquet theory, the factorization of fiber operators, the perturbation theory and the relation between effective masses for Laplacians on discrete and metric graphs, obtained in our paper.