Spectral invariants for discrete Schrödinger operators on periodic graphs

Abstract

The aim of this article is to present a complete system of Floquet spectral invariants for the discrete Schrödinger operators with periodic potentials on periodic graphs. These invariants are polynomials in the potential and determined by cycles in the quotient graph from some specific cycle sets. We discuss some properties of these invariants and give an explicit expression for the linear and quadratic (in the potential) Floquet spectral invariants. The constructed system of spectral invariants can be used to study the sets of isospectral periodic potentials for the Schrödinger operators on periodic graphs. In particular, we deduce that under certain assumptions, if a real potential is isospectral to the zero (respectively, “degree”) potential, then it must be the zero (respectively, “degree”) potential.