Cantor spectrum for the almost Mathieu equation. (English) Zbl 0516.47018

For a dense \(G_\delta\) of pairs \((\lambda, \alpha)\) in \(\mathbb R^2\), the authors prove that the operator \[ (Hu)(n)=u(n+1)+u(n-1)+\lambda\,\cos(2\pi\alpha n+\theta)u(n) \] has a nowhere dense spectrum. Additional results are proved for rational \(\alpha\). For example, if \(\alpha=(2p+1)/2q\) and \(\theta=0\) [resp. \(\pi/2q\)] if \(q\) is even [resp. odd] then one spectral gap is closed for all \(\lambda\). The paper concludes with a conjecture about how the main theorem might be strengthened using \(K\)-theory or homotopy.
Reviewer: A. L. Andrew


47B39 Linear difference operators
47A10 Spectrum, resolvent
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