95-355 Last Y.

Almost Everything about the Almost Mathieu Operator I (22K, TeX) Jul 5, 95

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Abstract. We review some aspects of the spectral theory of the Almost Mathieu operator (acting on $\ell^2(Z)$): $$(H_{\alpha, \lambda, \theta} u)(n) = u(n+1)+u(n-1)+\lambda\cos (2\pi\alpha n+\theta)u(n)\;,$$ where $\alpha,\lambda,\theta \in R$. We concentrate on the spectrum as a set, and describe the partial results obtained so far on the conjectures that, for irrational $\alpha$, it is a Cantor set and has Lebesgue measure $|4-2|\lambda||$.

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