Spectral band bracketing for Laplacians on periodic metric graphs

TL;DR

This paper introduces a method to localize spectral bands of Laplacians on periodic metric graphs by relating them to eigenvalues of Dirichlet and Neumann operators, enhancing understanding of their spectral structure.

Contribution

It provides a novel spectral band localization technique for Laplacians on periodic metric graphs using Dirichlet and Neumann eigenvalues, connecting discrete and metric spectra.

Findings

Spectral bands are localized via Dirichlet and Neumann eigenvalues.

The approach links spectra of discrete and metric Laplacians.

Results improve understanding of spectral structure on periodic graphs.

Abstract

We consider Laplacians on periodic metric graphs with unit-length edges. The spectrum of these operators 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. Our main result is a localization of spectral bands in terms of eigenvalues of Dirichlet and Neumann operators on a fundamental domain of the periodic graph. The proof is based on the spectral band localization for discrete Laplacians and on the relation between the spectra of discrete and metric Laplacians.