Trace formulas for magnetic Schr\"odinger operators on periodic graphs and their applications

TL;DR

This paper derives trace formulas for magnetic Schr"odinger operators on periodic graphs, linking spectral properties to geometric and magnetic parameters, and provides sharp lower bounds for the total bandwidth.

Contribution

It introduces explicit trace formulas for these operators, connecting spectral data with graph geometry and magnetic fluxes, and establishes sharp bandwidth estimates.

Findings

Trace formulas expressed as finite Fourier series of quasimomentum.

Lower bounds for total bandwidth depending on geometric and magnetic parameters.

Sharpness of the bandwidth estimates demonstrated.

Abstract

We consider Schr\"odinger operators with periodic magnetic and electric potentials on periodic discrete graphs. The spectrum of such operators consists of a finite number of bands. We determine trace formulas for the magnetic Schr\"odinger operators. The traces of the fiber operators are expressed as finite Fourier series of the quasimomentum. The coefficients of the Fourier series are given in terms of the magnetic fluxes, electric potentials and cycles in the quotient graph from some specific cycle sets. Using the trace formulas we obtain new lower estimates of the total bandwidth for the magnetic Schr\"odinger operator in terms of geometric parameters of the graph, magnetic fluxes and electric potentials. We show that these estimates are sharp.