Trace formulas for Schr\"odinger operators on periodic graphs

TL;DR

This paper derives trace formulas for Schr"odinger operators with periodic potentials on discrete graphs, expressing spectral properties in terms of Fourier series related to graph cycles, potentials, and quasimomentum.

Contribution

It introduces new trace formulas for these operators, linking spectral data to graph cycles and potentials through Fourier series representations.

Findings

Trace formulas for Schr"odinger operators on periodic graphs.

Explicit Fourier series expressions for traces of fiber operators.

Trace formulas for heat kernel, resolvent, and determinants.

Abstract

We consider Schr\"odinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We determine trace formulas for the Schr\"odinger operators. The proof is based on the decomposition of the Schr\"odinger operators into a direct integral and a specific representation of fiber 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 potentials and cycles in the quotient graph from some specific cycle sets. We also present the trace formulas for the heat kernel and the resolvent of the Schr\"odinger operators and the determinant formulas.