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
Full Text: DOI


[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
[3] {\scJ. Avron and B. Simon}, Almost periodic Schrödinger operators. II. The density of states, Caltech preprint.
[4] {\scJ. Bellissard, R. Lima, and D. Testard}, Almost random operators: K-theory and spectral properties, in preparation; see also {\scJ. bellissard}, Schrödinger operators with almost periodic potentials: An overview, in “Springer Lecture Notes in Physics, 153.”
[5] {\scV. Chulaevsky}, Russian Math. surveys, in press.
[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
[7] {\scG. Elliott}, Gaps in the spectrum of an almost periodic Schrödinger operator, Warwick preprint.
[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)
[10] {\scR. Johnson}, The recurrent Hill’s equation, J. Differential Equations, in press. · Zbl 0535.34021
[11] {\scR. Johnson}, private communication.
[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, () · Zbl 0517.47006
[16] {\scB. Simon}, Almost periodic Schrödinger operators, a review, Adv. Appl. Math., in press.
[17] {\scD. Thouless}, private communication; {\scG. Andre and S. Aubry}, private communication.
[18] {\scM. Herman}, A method for minorizing the Lyaponov exponent and several examples showing the local character of a theorem of Arnold and Moser on the torus of dimension 2, École Polytechnique preprint.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.