There is a ton of material available about this operator and its properties; this page is just a suggestion of where to get started.

D.R. Hofstader's paper on "Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields" is a good beginning, especially concerning the physical origins, in Physical Review B, Vol. 14 (1976), 2239-2249.

For results concerning the Cantor-set structure of the spectra, try the article by M.D. Choi, G.A. Elliott and N. Yui in Inventiones Mathematica, Vol. 99 (1990), 225-246 called "Gauss polynomials and the rotation algebra," or J. Bellissard and B. Simon's paper in the JFA Vol. 48 (1982), 408-419, "Cantor spectrum for the almost Mathieu Equation."

For some details on how to quickly compute the spectrum of these things, you might check out my paper "Reflections on the almost Mathieu Operator", which will appear in the Journal of Integral Equations and Operators, and also available on the usual preprint servers (and on my home page soon!)

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