On the spectrums of ergodic Schrodinger operators with finitely valued potentials

TL;DR

This paper investigates how the spectrum measure of ergodic Schrödinger operators with finitely valued potentials diminishes as the coupling constant increases, providing bounds and conditions for this behavior.

Contribution

It establishes that the spectrum measure tends to zero with increasing coupling and identifies necessary conditions for this phenomenon in ergodic operators.

Findings

Spectrum measure tends to zero as coupling increases

Provides quantitative upper bounds for spectrum measure

Identifies necessary recurrence conditions for the results

Abstract

We show that the measure of the spectrum of Schr\"odinger operator with potential defined by non-constant function over any minimal aperiodic finite subshift tends to zero, as the coupling constant tends to infinity. We also obtained a quantitative upper bound for the measure of the spectrum. This follows from a result we proved for ergodic Schr\"odinger operators with finitely valued potentials under two conditions on the recurrence property of the shift. We also show that one of these conditions is necessary for such result.