Spectral transition line for the extended Harper's model in the positive Lyapunov exponent regime

TL;DR

This paper investigates the spectral transition line of the extended Harper's model in the positive Lyapunov exponent regime, revealing the coexistence of pure point and singular continuous spectra for dense frequency subsets.

Contribution

It demonstrates the occurrence of both pure point and singular continuous spectra on the transition line for dense frequency sets in the extended Harper's model.

Findings

Pure point spectrum occurs on dense frequency subsets.

Singular continuous spectrum also occurs densely.

Spectral transition line contains both spectral types.

Abstract

We study the spectral transition line of the extended Harper's model in the positive Lyapunov exponent regime. We show that both pure point spectrum and purely singular continuous spectrum occur for dense subsets of frequencies on the transition line.