The absolutely continuous spectrum of the almost Mathieu operator

TL;DR

This paper proves that the spectrum of the almost Mathieu operator is absolutely continuous precisely when the coupling is subcritical, resolving a major open problem in Schrödinger operator theory.

Contribution

It establishes a complete characterization of the spectrum type for the almost Mathieu operator based on the coupling strength, solving a longstanding problem.

Findings

Spectrum is absolutely continuous in the subcritical regime

Spectrum is not absolutely continuous in the supercritical regime

Addresses Problem 6 from Barry Simon's list

Abstract

We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon's list of Schr\"odinger operator problems for the twenty-first century.