Bellissard, J.; Simon, B. Cantor spectrum for the almost Mathieu equation. (English) Zbl 0516.47018 J. Funct. Anal. 48, No. 3, 408-419 (1982). 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 Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 72 Documents MSC: 47B39 Linear difference operators 47A10 Spectrum, resolvent Keywords:spectral gap; \(K\)-theory; homotopy PDFBibTeX XMLCite \textit{J. Bellissard} and \textit{B. Simon}, J. Funct. Anal. 48, 408--419 (1982; Zbl 0516.47018) Full Text: DOI References: [1] Avron, J.; Simon, B., The asymptotics of the gap in the Mathieu equation, Ann. Phys., 134, 76-84 (1981) · Zbl 0464.34020 [2] Avron, J.; Simon, B., Almost periodic Schrödinger operators. I. Limit periodic potentials, Commun. Math. Phys., 82, 101-120 (1981) · Zbl 0484.35069 [6] Dubrovin, B.; Matveev, V.; Novikov, S., Nonlinear equations of KdV type, finite zone linear operators and abelian varieties, Russ. Math. Surveys, 31, 59-146 (1976) · Zbl 0346.35025 [8] Harrell, E., Double wells, Commun. Math. Phys., 75, 239-261 (1980) · Zbl 0445.35036 [9] Hofstader, D., Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B, 14, 2239-2249 (1976) [12] Johnson, R.; Moser, J., The rotation number for almost periodic potentials, Commun. Math. Phys., 84, 403-438 (1982) · Zbl 0497.35026 [13] Moser, J., An example of a Schrödinger operator with almost periodic potential and nowhere dense spectrum, Comment. Math. Helv., 56, 198-224 (1981) · Zbl 0477.34018 [14] Pimsner, M.; Voiculescu, D., Imbedding the irrational rotation \(C^∗\)-algebra in an AF-algebra, J. Operator Th., 4, 201-218 (1980) · Zbl 0525.46031 [15] Reed, M.; Simon, B., Methods of Modern Mathematical Physics, (Analysis of Operators, Vol. IV (1978), Academic Press: Academic Press New York) · Zbl 0517.47006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.