Norm Estimates of Almost Mathieu Operators

where θ, λ and φ are real numbers, and (ξn)n denotes the canonical orthonormal basis in l2. The study of the spectral properties of this class of operators has attracted a significant amount of interest in the past couple of decades (see [5], [3], [8], [1], [10], [14], [12], [7] for some of the most important developments). Most of this work has focused on the “Ten Martini problem” of M. Kac, concerning the possible values of the labels of the gaps that appear in the spectrum of these kinds of operators, and on the localization properties of the spectrum. The almost Mathieu operator H(θ, λ, φ) can be regarded as the image of the self-adjoint element Hθ,λ = U+U ∗+(λ/2)(V +V ∗) in the representation of the rotation C∗-algebra Aθ = C ∗(U, V unitaries ; UV = e2πiθV U) that maps U to the bilateral shift u0 defined on l 2 by u0ξn = ξn−1, and V to the diagonal unitary v0 defined by v0ξn = e ξn. The operator Hθ = Hθ,2 is called a Harper operator. When θ = p/q is rational with 0 ≤ p < q coprime integers, the spectrum of Hθ,λ, viewed as an element of Aθ, consists either in the union of q disjoint intervals (when q is odd), or of q−1 disjoint intervals (when q is even). This is best illustrated in Hofstadter’s butterfly ([11]) in Figure 1.