Cantor spectrum for the almost Mathieu equation
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.
scientific article; zbMATH DE number 3816429