Second Phase transition line

TL;DR

This paper investigates the phase transition line of the almost Mathieu operator, demonstrating that both singular continuous and pure point spectra occur densely across different frequencies, revealing complex spectral behavior.

Contribution

It establishes the density of frequencies leading to singular continuous and pure point spectra, advancing understanding of spectral phases in the almost Mathieu operator.

Findings

Both spectral types occur densely across frequencies.

The phase transition line separates different spectral regimes.

Spectral behavior is more intricate than previously understood.

Abstract

We study the phase transion line of the almost Mathieu operator, that separates arithmetic regions corresponding to singular continuous and a.e. pure point regimes, and prove that both purely singular continuous and a.e. pure point spectrum occur for dense sets of frequencies.