Two-sided estimates of total bandwidth for Schr\"odinger operators on periodic graphs

TL;DR

This paper derives precise two-sided estimates for the total bandwidth of Schrödinger operators on periodic graphs, linking spectral properties to geometric and potential parameters, and demonstrates the sharpness of these bounds.

Contribution

It provides sharp, two-sided estimates of the total bandwidth for Schrödinger operators on periodic graphs, expressed through geometric and potential parameters, using Floquet theory and trace formulas.

Findings

Estimates are sharp and become equalities for specific graphs and potentials.

Bandwidth bounds are expressed via Fourier coefficients related to graph cycles.

The approach uses Floquet theory and trace formulas to connect spectral and geometric data.

Abstract

We consider Schr\"odinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We obtain two-sided estimates of the total bandwidth for the Schr\"odinger operators in terms of geometric parameters of the graph and the potentials. In particular, we show that these estimates are sharp. It means that these estimates become identities for specific graphs and potentials. The proof is based on the Floquet theory and trace formulas for fiber operators. The traces are expressed as finite Fourier series of the quasimomentum with coefficients depending on the potentials and cycles of the quotient graph from some specific cycle sets. In order to obtain our results we estimate these Fourier coefficients in terms of geometric parameters of the graph and the potentials.