On continuous spectrum of magnetic Schr\"odinger operators on periodic discrete graphs

TL;DR

This paper investigates the spectral properties of magnetic Schr"odinger operators on periodic discrete graphs, revealing conditions for the absence of absolutely continuous spectrum and how the spectrum varies with magnetic potential strength.

Contribution

It provides necessary and sufficient conditions for the absence of absolutely continuous spectrum and analyzes how the spectrum changes with magnetic potential coupling.

Findings

Absolutely continuous spectrum can be empty for certain graphs and magnetic fields.

Conditions are established for when the a.c. spectrum is absent.

The spectrum generally has an a.c. component for most values of the magnetic potential.

Abstract

We consider Schr\"odinger operators with periodic electric and magnetic potentials on periodic discrete graphs. The spectrum of such operators consists of an absolutely continuous (a.c.) part (a union of a finite number of non-degenerate bands) and a finite number of eigenvalues of infinite multiplicity. We prove the following results: 1) the a.c. spectrum of the magnetic Schr\"odinger operators is empty for specific graphs and magnetic fields; 2) we obtain necessary and sufficient conditions under which the a.c. spectrum of the magnetic Schr\"odinger operators is empty; 3) the spectrum of the magnetic Schr\"odinger operator with each magnetic potential $t\alpha$, where $t$ is a coupling constant, has an a.c. component for all except finitely many $t$ from any bounded interval.