Abstract
We study the phase transion line of the almost Mathieu operator, that separates arithmetic regions corresponding to singular continuous and a.e. pure point regimes, and prove that both purely singular continuous and a.e. pure point spectrum occur for dense sets of frequencies.
Similar content being viewed by others
Notes
References
Aubry, S., André, G.: Analyticity breaking and Anderson localization in incommensurate lattices. In: Group Theoretical Methods in Physics. Proceedings of Eighth International Colloquium Kiryat Anavim, 1979, Hilger, Bristol, pp. 133–164 (1980)
Avila, A.: Absolutely continuous spectrum for the almost Mathieu operator with subcritical coupling. http://w3.impa.br/~avila/
Avila, A.: Global theory of one-frequency Schrödinger operators. Acta Math. 215, 1–54 (2015)
Avila, A., Damanik, D.: Absolute continuity of the integrated density of states for the almost Mathieu operator with non-critical coupling. Invent. Math. 172, 439–453 (2008)
Avila, A., Fayad, B., Krikorian, R.: A KAM scheme for \({\rm SL}(2,\mathbb{R})\) cocycles with Liouvillean frequencies. Geom. Funct. Anal. 21, 1001–1019 (2011)
Avila, A., Jitomirskaya, S.: Almost localization and almost reducibility. J. Eur. Math. Soc. 12, 93–131 (2010)
Avila, A., Jitomirskaya, S., Marx, C.A. : Spectral theory of Extended Harper’s Model and a question by Erdös and Szekeres. arXiv:1602.05111
Avila, A., Krikorian, R.: Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. Math. 164, 911–940 (2006)
Avila, A., You, J., Zhou, Q.: Sharp phase transitions for the almost Mathieu operator. Duke Math. J. (Accepted)
Avron, J., Mouche, P.V., Simon, B.: On the measure of the spectrum for the almost Mathieu operator. Commun. Math. Phys. 132(1), 103–118 (1990)
Bourgain, J., Jitomirskaya, S.: Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential. J. Stat. Phys. 108, 1203–1218 (2002)
Gordon, A.Y.: The point spectrum of one-dimensional Schrödinger operator. Uspehi Mat. Nauk. 31, 257–258 (1976). (Russian)
Herman, M.: Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol’d et de Moser sur le tore de dimension \(2\). Comment. Math. Helv. 58(3), 453–502 (1983)
Hou, X., You, J.: Almost reducibility and non-perturbative reducibility of quasiperiodic linear systems. Invent. Math. 190, 209–260 (2012)
Jitomirskaya, S.: Almost everything about the almost mathieu operator, II. In: Proceedings of XI International Congress of Mathematical Physics, pp. 373–382. Int. Press (1995)
Jitomirskaya, S.: Metal-insulator transition for the almost Mathieu operator. Ann. Math. (2) 150(3), 1159–1175 (1999)
Jitomirskaya, S.: Ergodic Schrödinger operator (on one foot). In: Proceedings of Symposia in Pure Mathematics, vol. 76, no. 2, pp. 613–647 (2007)
Jitomirskaya, S., Kachkovskiy, I.: \(L^2\) reducibility and Localization for quasiperiodic operators. Math. Res. Lett. 23(2), 431–444 (2016)
Jitomirskaya, S., Liu, W.: Universal hierarchical structure of quasiperiodic eigenfunctions. arXiv:1609.08664
Jitomirskaya, S., Liu, W.: Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transitions in phase. (In preparation)
Jitomirskaya, S., Simon, B.: Operators with singular continuous spectrum. III. Almost periodic Schrödinger operators. Commun. Math. Phys. 165(1), 201–205 (1994)
Jitomirskaya, S., Yang, F.: Singular continuous spectrum for singular potentials. Commun. Math. Phys. 351(3), 1127–1135 (2017)
Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84(3), 403–438 (1982)
Kotani, S.: Lyaponov indices determine absolutely continuous spectra of stationary random onedimensional Schrondinger operators. In: Ito, K. (ed.) Stochastic Analysis, pp. 225–248. North Holland, Amsterdam (1984)
Last, Y.: A relation between absolutely continuous spectrum of ergodic Jacobi matrices and the spectra of periodic approximants. Commun. Math. Phys. 151, 183–192 (1993)
Last, Y.: Zero measure spectrum for the almost Mathieu operator. Commun. Math. Phys. 164, 421–432 (1994)
You, J., Zhou, Q.: Embedding of analytic quasi-periodic cocycles into analytic quasi-periodic linear systems and its applications. Commun. Math. Phys. 323, 975–1005 (2013)
Acknowledgements
A.A. and Q.Z. were partially supported by the ERC Starting Grant “Quasiperiodic”. S.J. was a 2014–15 Simons Fellow, and was partially supported by NSF DMS-1401204. Q.Z. was also supported by “Deng Feng Scholar Program B” of Nanjing University and NNSF of China (11671192), he would like to thank the hospitality of the UCI where this work was started. S.J. and Q.Z. are grateful to the Isaac Newton Institute for Mathematical Sciences, Cambridge, for its hospitality supported by EPSRC Grant No. EP/K032208/1, during the programme “Periodic and Ergodic Spectral Problems” where they worked on this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Avila, A., Jitomirskaya, S. & Zhou, Q. Second phase transition line. Math. Ann. 370, 271–285 (2018). https://doi.org/10.1007/s00208-017-1543-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-017-1543-1