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 so-called almost Mathieu equation, 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.
- Publication:
-
Progress of Theoretical Physics Supplement
- Pub Date:
- 1999
- DOI:
- 10.1143/PTPS.134.171
- arXiv:
- arXiv:hep-th/9909083
- Bibcode:
- 1999PThPS.134..171W
- Keywords:
-
- High Energy Physics - Theory;
- Astrophysics;
- Condensed Matter - Strongly Correlated Electrons;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- Exactly Solvable and Integrable Systems
- E-Print:
- 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