Schr\"odinger operators on periodic discrete graphs

TL;DR

This paper analyzes the spectral properties of Schr"odinger operators with periodic potentials on discrete graphs, providing estimates, stability results, and insights into flat band phenomena using Floquet theory.

Contribution

It introduces new spectral estimates and stability results for Schr"odinger operators on periodic graphs, with explicit characterization of flat bands and geometric dependencies.

Findings

Spectral measure estimates in terms of graph geometry

Existence and localization of flat bands in specific graphs

Stability estimates for the spectrum under perturbations

Abstract

We consider Schr\"odinger operators with periodic potentials on periodic discrete graphs. The spectrum of the Schr\"odinger operator consists of an absolutely continuous part (a union of a finite number of non-degenerated bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We obtain estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graph and show that they become identities for some class of graphs. Moreover, we obtain stability estimates and show the existence and positions of large number of flat bands for specific graphs. The proof is based on the Floquet theory and the precise representation of fiber Schr\"odinger operators, constructed in the paper.