Absolute Continuity of the Integrated Density of States for the Almost Mathieu Operator with Non-Critical Coupling

TL;DR

This paper proves that the integrated density of states for the almost Mathieu operator is absolutely continuous when the coupling is non-critical, confirming a key case in spectral theory and solving a longstanding problem.

Contribution

It establishes the absolute continuity of the integrated density of states for non-critical coupling in the almost Mathieu operator, resolving a major open problem in spectral theory.

Findings

Absolute continuity holds for non-critical coupling

Spectrum is purely absolutely continuous for almost every phase in the subcritical case

Addresses Problem 6 from Barry Simon's list of Schrödinger operator problems

Abstract

We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measure-theoretical case of Problem 6 of Barry Simon's list of Schr\"odinger operator problems for the twenty-first century.