Trace formulas for magnetic Schrödinger operators on periodic graphs and their applications

Abstract

We consider Schrödinger operators with periodic magnetic and electric potentials on periodic discrete graphs. The spectrum of such operators consists of a finite number of bands. Some of them and even all may be degenerate. We determine trace formulas for the magnetic Schrödinger 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 Schrödinger operator in terms of geometric parameters of the graph, magnetic fluxes and electric potentials. We show that these estimates are sharp.