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

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.

Based on the lecture given at 13th Nishinomiya-Yukawa Memorial Symposium on Dynamics of Fields and Strings, Nishinomiya, Japan, 12-13 Nov 1998, and talk given at YITP Workshop on New Aspects of Strings and Fields, Kyoto, Japan, 16-18 Nov 1998

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• Critical States and Fractal Attractors in Fractal Tongues: Localization in the Harper map
• Fractal Structure of the Harper Map Phase Diagram from Topological Hierarchical Classification
• Landau Levels from the Bethe Ansatz Equations
• Landau problem with a general time-dependent electric field
• Phase Diagrams of the Harper Map and the Golden Staircase
• Metal-insulator transition for the almost Mathieu operator