On the singular spectrum of the Almost Mathieu operator. Arithmetics and Cantor spectra of integrable models

TL;DR

This paper reviews recent progress in understanding the spectrum of the Almost Mathieu operator, highlighting its singular continuous nature, hierarchical structure, and integrability, with insights gained from Bethe Ansatz analysis.

Contribution

It presents a comprehensive review of the spectral properties and integrability of the Almost Mathieu operator, including recent asymptotic solutions via Bethe Ansatz.

Findings

Spectrum is a pure singular continuum with hierarchical structure

Almost Mathieu operator is proven to be integrable

Asymptotic solutions obtained through Bethe Ansatz analysis

Abstract

I review a recent progress towards solution of the Almost Mathieu equation (A.G. Abanov, J.C. Talstra, P.B. Wiegmann, Nucl. Phys. B 525, 571, 1998), known also as Harper's equation or Azbel-Hofstadter problem. The spectrum of this equation is known to be a pure singular continuum with a rich hierarchical structure. Few years ago it has been found that the almost Mathieu operator is integrable. An asymptotic solution of this operator became possible due analysis the Bethe Ansatz equations.