Dry Ten Martini problem for the non-self-dual extended Harper’s model
HTML articles powered by AMS MathViewer
- by Rui Han PDF
- Trans. Amer. Math. Soc. 370 (2018), 197-217 Request permission
Abstract:
In this paper we prove the dry version of the Ten Martini problem: Cantor spectrum with all gaps open, for the extended Harper’s model in the non-self-dual region for Diophantine frequencies.References
- Serge Aubry and Gilles André, Analyticity breaking and Anderson localization in incommensurate lattices, Group theoretical methods in physics (Proc. Eighth Internat. Colloq., Kiryat Anavim, 1979) Ann. Israel Phys. Soc., vol. 3, Hilger, Bristol, 1980, pp. 133–164. MR 626837
- A. Avila, Absolutely continuous spectrum for the almost Mathieu operator, Preprint.
- A. Avila, Almost reducibility and absolute continuity I, Preprint.
- Artur Avila, Global theory of one-frequency Schrödinger operators, Acta Math. 215 (2015), no. 1, 1–54. MR 3413976, DOI 10.1007/s11511-015-0128-7
- Artur Avila and Svetlana Jitomirskaya, The Ten Martini Problem, Ann. of Math. (2) 170 (2009), no. 1, 303–342. MR 2521117, DOI 10.4007/annals.2009.170.303
- Artur Avila and Svetlana Jitomirskaya, Almost localization and almost reducibility, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 1, 93–131. MR 2578605, DOI 10.4171/JEMS/191
- A. Avila, S. Jitomirskaya, and C. Marx, Spectral theory of extended Harper’s model and a question by Erdős and Szekeres, Preprint.
- Joseph Avron and Barry Simon, Almost periodic Schrödinger operators. II. The integrated density of states, Duke Math. J. 50 (1983), no. 1, 369–391. MR 700145, DOI 10.1215/S0012-7094-83-05016-0
- A. Avila, J. You, and Q. Zhou, Dry Ten Martini problem in non-critical case, in preparation.
- Jean Bellissard, Ricardo Lima, and Daniel Testard, Almost periodic Schrödinger operators, Mathematics + physics. Vol. 1, World Sci. Publishing, Singapore, 1985, pp. 1–64. MR 849342, DOI 10.1142/9789814415125_{0}001
- Ju. M. Berezans′kiĭ, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. MR 0222718, DOI 10.1090/mmono/017
- Man Duen Choi, George A. Elliott, and Noriko Yui, Gauss polynomials and the rotation algebra, Invent. Math. 99 (1990), no. 2, 225–246. MR 1031901, DOI 10.1007/BF01234419
- S. Jitomirskaya, D. A. Koslover, and M. S. Schulteis, Localization for a family of one-dimensional quasiperiodic operators of magnetic origin, Ann. Henri Poincaré 6 (2005), no. 1, 103–124. MR 2121278, DOI 10.1007/s00023-005-0200-5
- S. Jitomirskaya and C. A. Marx, Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper’s model, Comm. Math. Phys. 316 (2012), no. 1, 237–267. MR 2989459, DOI 10.1007/s00220-012-1465-4
- R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys. 84 (1982), no. 3, 403–438. MR 667409, DOI 10.1007/BF01208484
- Joaquim Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys. 244 (2004), no. 2, 297–309. MR 2031032, DOI 10.1007/s00220-003-0977-3
- Barry Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. MR 670130, DOI 10.1090/S0273-0979-1982-15041-8
Additional Information
- Rui Han
- Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875
- MR Author ID: 1138295
- Email: rhan2@uci.edu
- Received by editor(s): March 9, 2016
- Published electronically: July 7, 2017
- Additional Notes: This research was partially supported by NSF grant DMS1401204
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 197-217
- MSC (2010): Primary 47B36; Secondary 39A70, 47B39, 81Q10, 47A10
- DOI: https://doi.org/10.1090/tran/6989
- MathSciNet review: 3717978