Exponential Decay of the lengths of Spectral Gaps for Extended Harper's Model with Liouvillean Frequency

TL;DR

This paper proves that in the extended Harper's model with Liouvillean frequency, the spectral gaps decrease exponentially in size, using reducibility results and averaging methods.

Contribution

It introduces a new approach combining quantitative reducibility and averaging to analyze spectral gap decay in a non-self dual model with Liouvillean frequency.

Findings

Spectral gaps decay exponentially in the extended Harper's model.

Established quantitative reducibility results for the model.

Applied averaging methods to prove spectral gap decay.

Abstract

In this paper, we study the non-self dual extended Harper's model with Liouvillean frequency. By establishing quantitative reducibility results together with the averaging method, we prove that the lengths of spectral gaps decay exponentially.