Sharp Phase transitions for the almost Mathieu operator

TL;DR

This paper precisely identifies the conditions under which the almost Mathieu operator transitions from singular continuous to pure point spectrum, resolving a longstanding conjecture and enhancing understanding of spectral phase transitions.

Contribution

It provides a sharp characterization of phase transitions in the almost Mathieu operator, confirming Jitomirskaya's conjecture and advancing spectral theory knowledge.

Findings

Locates the exact point of phase transition

Confirms Jitomirskaya's conjecture

Provides a complete description of spectral phase transitions

Abstract

It is known that the spectral type of the almost Mathieu operator depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place, which solves Jitomirskaya's conjecture in \cite{Ji95,J07}. Together with \cite{Aab}, we give the sharp description of phase transitions for the almost Mathieu operator.