zbMATH — the first resource for mathematics

The Ten Martini problem. (English) Zbl 1166.47031
From the Introduction: In this paper, we solve the Ten Martini Problem as stated in [B. Simon, in: Mathematical physics 2000. International congress, London, GB, 2000 (London: Imperial College Press), 283–288 (2000; Zbl 1074.81521)].
Theorem. The spectrum of the almost Mathieu operator is a Cantor set for all irrational \(\alpha\) and for all \(\lambda \neq 0\).
The almost Mathieu operator is the Schrödinger operator on \(\ell^2(\mathbb{Z})\),
\[ (H_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+2\lambda\cos 2\pi(\theta+n\alpha)u_n, \]
where \(\lambda,\alpha,\theta \in \mathbb{R}\) are parameters (called the coupling, frequency, and phase, respectively), and one assumes that \(\lambda\neq 0\). The interest in this particular model is motivated both by its connections to physics and by a remarkable richness of the related spectral theory. This has made the latter a subject of intense research in the last three decades.
If \(\alpha=\frac{p}{q}\) is rational, it is well-known that the spectrum consists of the union of \(q\) intervals called bands, possibly touching at the endpoints. In the case of irrational \(\alpha\), the spectrum \(\Sigma_{\lambda,\alpha}\) (which in this case does not depend on \(\theta\)) has been conjectured for a long time to be a Cantor set. To prove this conjecture has been dubbed The Ten Martini Problem by B. Simon [op. cit.]. For a history of this problem, see [Y. Last, in: Sturm–Liouville theory. Past and present (Basel: Birkhäuser), 99–120 (2005; Zbl 1098.39011)].

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47A10 Spectrum, resolvent
Full Text: DOI Link
[1] A. Avila and R. Krikorian, ”Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles,” Ann. of Math., vol. 164, iss. 3, pp. 911-940, 2006. · Zbl 1138.47033 · doi:10.4007/annals.2006.164.911 · euclid:annm/1172614615
[2] A. Avila and R. Krikorian, ”Quasiperiodic \(\operatorname{SL}(2,\mathbbR)\) cocycles,” , in preparation.
[3] J. Avron, P. H. M. van Mouche, and B. Simon, ”On the measure of the spectrum for the almost Mathieu operator,” Comm. Math. Phys., vol. 132, iss. 1, pp. 103-118, 1990. · Zbl 0724.47002 · doi:10.1007/BF02278001 · projecteuclid.org
[4] J. Avron and B. Simon, ”Almost periodic Schrödinger operators. II: The integrated density of states,” Duke Math. J., vol. 50, iss. 1, pp. 369-391, 1983. · Zbl 0544.35030 · doi:10.1215/S0012-7094-83-05016-0
[5] Y. M. Azbel, ”Energy spectrum of a conduction electron in a magnetic field,” Sov. Phys. JETP, vol. 19, pp. 634-645, 1964.
[6] J. M. Berezanscprimekiui, Expansions in Eigenfunctions of Selfadjoint Operators, Providence, R.I.: Amer. Math. Soc., 1968.
[7] J. Bourgain and S. Jitomirskaya, ”Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential,” J. Statist. Phys., vol. 108, iss. 5-6, pp. 1203-1218, 2002. · Zbl 1039.81019 · doi:10.1023/A:1019751801035 · arxiv:math-ph/0110040
[8] J. Béllissard and B. Simon, ”Cantor spectrum for the almost Mathieu equation,” J. Funct. Anal., vol. 48, iss. 3, pp. 408-419, 1982. · Zbl 0516.47018 · doi:10.1016/0022-1236(82)90094-5
[9] M. D. Choi, G. A. Elliott, and N. Yui, ”Gauss polynomials and the rotation algebra,” Invent. Math., vol. 99, iss. 2, pp. 225-246, 1990. · Zbl 0665.46051 · doi:10.1007/BF01234419 · eudml:143758
[10] A. Furman, ”On the multiplicative ergodic theorem for uniquely ergodic systems,” Ann. Inst. H. Poincaré Probab. Statist., vol. 33, iss. 6, pp. 797-815, 1997. · Zbl 0892.60011 · doi:10.1016/S0246-0203(97)80113-6 · numdam:AIHPB_1997__33_6_797_0 · eudml:77590
[11] B. Helffer and J. Sjöstrand, ”Semiclassical analysis for Harper’s equation. III: Cantor structure of the spectrum,” Mém. Soc. Math. France, iss. 39, pp. 1-124, 1989. · Zbl 0699.35273 · numdam:SEDP_1988-1989____A2_0 · eudml:111971
[12] M. Herman, ”Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol\cprime d et de Moser sur le tore de dimension \(2\),” Comment. Math. Helv., vol. 58, iss. 3, pp. 453-502, 1983. · Zbl 0554.58034 · doi:10.1007/BF02564647 · eudml:139950
[13] L. Hörmander, Notions of Convexity, Boston: Birkhäuser, 1994. · Zbl 1083.32033
[14] S. Y. Jitomirskaya, ”Metal-insulator transition for the almost Mathieu operator,” Ann. of Math., vol. 150, iss. 3, pp. 1159-1175, 1999. · Zbl 0946.47018 · doi:10.2307/121066 · www.math.princeton.edu · eudml:120545 · arxiv:math/9911265
[15] S. Jitomirskaya, D. A. Koslover, and M. S. Schulteis, ”Localization for a family of one-dimensional quasiperiodic operators of magnetic origin,” Ann. Henri Poincaré, vol. 6, iss. 1, pp. 103-124, 2005. · Zbl 1062.81029 · doi:10.1007/s00023-005-0200-5
[16] Y. S. Jitomirskaya and I. V. Krasovsky, ”Continuity of the measure of the spectrum for discrete quasiperiodic operators,” Math. Res. Lett., vol. 9, iss. 4, pp. 413-421, 2002. · Zbl 1020.47002 · doi:10.4310/MRL.2002.v9.n4.a1
[17] R. Johnson and J. Moser, ”The rotation number for almost periodic potentials,” Comm. Math. Phys., vol. 84, iss. 3, pp. 403-438, 1982. · Zbl 0497.35026 · doi:10.1007/BF01208484 · projecteuclid.org
[18] Y. Last, ”Zero measure spectrum for the almost Mathieu operator,” Comm. Math. Phys., vol. 164, iss. 2, pp. 421-432, 1994. · Zbl 0814.11040 · doi:10.1007/BF02101708 · projecteuclid.org
[19] Y. Last, ”Spectral theory of Sturm-Liouville operators on infinite intervals: A review of recent developments,” in Sturm-Liouville Theory, Amrein, W. O., Hinz, A. M., and Pearson, D. B., Eds., Basel: Birkhäuser, 2005, pp. 99-120. · Zbl 1098.39011
[20] J. Puig, ”Cantor spectrum for the almost Mathieu operator,” Comm. Math. Phys., vol. 244, iss. 2, pp. 297-309, 2004. · Zbl 1075.39021 · doi:10.1007/s00220-003-0977-3
[21] B. Simon, ”Kotani theory for one-dimensional stochastic Jacobi matrices,” Comm. Math. Phys., vol. 89, iss. 2, pp. 227-234, 1983. · Zbl 0534.60057 · doi:10.1007/BF01211829 · projecteuclid.org
[22] B. Simon, ”Schrödinger operators in the twenty-first century,” in Mathematical Physics 2000, Fokas, A., Grigoryan, A., Kibble, T., and Zegarlinski, B., Eds., London: Imp. Coll. Press, 2000, pp. 283-288. · Zbl 1074.81521
[23] Y. G. Sinaui, ”Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential,” J. Statist. Phys., vol. 46, iss. 5-6, pp. 861-909, 1987. · Zbl 0682.34023 · doi:10.1007/BF01011146
[24] P. van Mouche, ”The coexistence problem for the discrete Mathieu operator,” Comm. Math. Phys., vol. 122, iss. 1, pp. 23-33, 1989. · Zbl 0669.34016 · doi:10.1007/BF01221406 · projecteuclid.org
[25] C. De Concini and R. A. Johnson, ”The algebraic-geometric AKNS potentials,” Ergodic Theory Dynam. Systems, vol. 7, iss. 1, pp. 1-24, 1987. · Zbl 0636.35077 · doi:10.1017/S0143385700003783
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.