Magnetic Schr\"odinger operators on periodic discrete graphs

TL;DR

This paper analyzes the spectral properties of magnetic Schr"odinger operators on periodic discrete graphs, providing estimates for the spectrum's measure and its variation under magnetic perturbations using Floquet theory.

Contribution

It introduces new spectral estimates for magnetic Schr"odinger operators on periodic graphs and characterizes the spectrum's structure with respect to magnetic fluxes.

Findings

Spectrum consists of absolutely continuous bands and flat eigenvalues.

Lebesgue measure of spectrum estimated via Betti numbers.

Spectrum variation under magnetic perturbations quantified.

Abstract

We consider magnetic Schr\"odinger operators with periodic magnetic and electric potentials on periodic discrete graphs. The spectrum of the operators consists of an absolutely continuous part (a union of a finite number of non-degenerate bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We estimate the Lebesgue measure of the spectrum in terms of the Betti numbers and show that these estimates become identities for specific graphs. We estimate a variation of the spectrum of the Schr\"odinger operators under a perturbation by a magnetic field in terms of magnetic fluxes. The proof is based on Floquet theory and a precise representation of fiber magnetic Schr\"odinger operators constructed in the paper.