# Spectral estimates for Schr\"odinger operators on periodic discrete graphs

## TL;DR

This paper provides spectral estimates for Schr"odinger operators on periodic discrete graphs, linking spectral properties to geometric graph parameters and establishing bounds on spectral band lengths and effective masses.

## Contribution

It introduces new bounds and identities for the spectrum of Schr"odinger operators on periodic graphs, connecting spectral features with geometric graph parameters.

## Key findings

- Spectral measure estimates depend on graph geometry.
- First spectral band of Schr"odinger operators is non-degenerate.
- Two-sided bounds on spectral band length and effective mass.

## Abstract

We consider normalized Laplacians and their perturbations by periodic potentials (Schr\"odinger operators) 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) and 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 graphs and show that they become identities for some class of graphs. We determine two-sided estimates on the length of the first spectral band and on the effective mass at the bottom of the spectrum of the Laplace and Schr\"odinger operators. In particular, these estimates yield that the first spectral band of Schr\"odinger operators is non-degenerate.