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Almost Mathieu operators with completely resonant phases

  • WENCAI LIU (a1)


Let $\unicode[STIX]{x1D6FC}\in \mathbb{R}\backslash \mathbb{Q}$ and $\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})=\limsup _{n\rightarrow \infty }(\ln q_{n+1})/q_{n}<\infty$ , where $p_{n}/q_{n}$ is the continued fraction approximation to $\unicode[STIX]{x1D6FC}$ . Let $(H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}u)(n)=u(n+1)+u(n-1)+2\unicode[STIX]{x1D706}\cos 2\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}+n\unicode[STIX]{x1D6FC})u(n)$ be the almost Mathieu operator on $\ell ^{2}(\mathbb{Z})$ , where $\unicode[STIX]{x1D706},\unicode[STIX]{x1D703}\in \mathbb{R}$ . Avila and Jitomirskaya [The ten Martini problem. Ann. of Math. (2), 170(1) (2009), 303–342] conjectured that, for $2\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6FC}\mathbb{Z}+\mathbb{Z}$ , $H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}$ satisfies Anderson localization if $|\unicode[STIX]{x1D706}|>e^{2\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$ . In this paper, we develop a method to treat simultaneous frequency and phase resonances and obtain that, for $2\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6FC}\mathbb{Z}+\mathbb{Z}$ , $H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}$ satisfies Anderson localization if $|\unicode[STIX]{x1D706}|>e^{3\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$ .



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[1]Avila, A. and Jitomirskaya, S.. Solving the ten Martini problem. Mathematical Physics of Quantum Mechanics (Lecture Notes in Physics, 690). Springer, Berlin, 2006, pp. 516.
[2]Avila, A. and Jitomirskaya, S.. The ten Martini problem. Ann. of Math. (2) 170(1) (2009), 303342.
[3]Avila, A. and Jitomirskaya, S.. Almost localization and almost reducibility. J. Eur. Math. Soc. 12(1) (2010), 93131.
[4]Avila, A., Jitomirskaya, S. and Zhou, Q.. Second phase transition line. Math. Ann. 370(1–2) (2018), 271285.
[5]Avila, A., You, J. and Zhou, Q.. Dry ten Martini problem in non-critical case. Preprint.
[6]Avila, A., You, J. and Zhou, Q.. Sharp phase transitions for the almost Mathieu operator. Duke Math. J. 166(14) (2017), 26972718.
[7]Bellissard, J., Lima, R. and Testard, D.. Almost periodic Schrödinger operators. Mathematics+ Physics (Lectures on Recent Results, 1). World Scientific, Singapore, 1985, pp. 164.
[8]Berezanskii, J. M.. Expansions in Eigenfunctions of Self-Adjoint Operators (Translations of Mathematical Monographs, 17). American Mathematical Society, Providence, RI, 1968.
[9]Bourgain, J.. Green’s Function Estimates for Lattice Schrödinger Operators and Applications (Annals of Mathematics Studies, 158). Princeton University Press, Princeton, NJ, 2005.
[10]Bourgain, J. and Goldstein, M.. On nonperturbative localization with quasi-periodic potential. Ann. Math. 152(3) (2000), 835879.
[11]Bourgain, J., Goldstein, M. and Schlag, W.. Anderson localization for Schrödinger operators on Z2 with quasi-periodic potential. Acta Math. 188(1) (2002), 4186.
[12]Bourgain, J. and Jitomirskaya, S.. Absolutely continuous spectrum for 1D quasiperiodic operators. Invent. Math. 148(3) (2002), 453463.
[13]Chulaevsky, V. and Delyon, F.. Purely absolutely continuous spectrum for almost Mathieu operators. J. Statist. Phys. 55(5–6) (1989), 12791284.
[14]Dinaburg, E. and Sinai, Y. G.. The one-dimensional Schrödinger equation with a quasiperiodic potential. Funct. Anal. Appl. 9(4) (1975), 279289.
[15]Eliasson, L. H.. Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146(3) (1992), 447482.
[16]Fröhlich, J., Spencer, T. and Wittwer, P.. Localization for a class of one dimensional quasi-periodic Schrödinger operators. Commun. Math. Phys. 132(1) (1990), 525.
[17]Gordon, A. Y.. The point spectrum of the one-dimensional Schrödinger operator. Uspekhi Mat. Nauk 31(4) (1976), 257258.
[18]Gordon, A. Y., Jitomirskaya, S., Last, Y. and Simon, B.. Duality and singular continuous spectrum in the almost Mathieu equation. Acta Math. 178(2) (1997), 169183.
[19]Hadj Amor, S.. Hölder continuity of the rotation number for quasi-periodic co-cycles in SL(2, ℝ). Commun. Math. Phys. 287(2) (2009), 565588.
[20]Han, R.. Dry ten Martini problem for the non-self-dual extended Harper’s model. Trans. Amer. Math. Soc. 370(1) (2018), 197217.
[21]Han, R. and Jitomirskaya, S.. Full measure reducibility and localization for quasiperiodic Jacobi operators: a topological criterion. Adv. Math. 319 (2017), 224250.
[22]Jitomirskaya, S.. Almost everything about the almost Mathieu operator. II. XIth International Congress of Mathematical Physics (Paris, 1994). International Press, Cambridge, MA, 1995, pp. 373382.
[23]Jitomirskaya, S.. Metal-insulator transition for the almost Mathieu operator. Ann. Math. 150(3) (1999), 11591175.
[24]Jitomirskaya, S. and Kachkovskiy, I.. L 2 -reducibility and localization for quasiperiodic operators. Math. Res. Lett. 23(2) (2016), 431444.
[25]Jitomirskaya, S., Koslover, D. A. and Schulteis, M. S.. Localization for a family of one-dimensional quasiperiodic operators of magnetic origin. Ann. Henri Poincaré 6(1) (2005), 103124.
[26]Jitomirskaya, S. and Liu, W.. Arithmetic spectral transitions for the Maryland model. Commun. Pure Appl. Math. 70(6) (2017), 10251051.
[27]Jitomirskaya, S. and Liu, W.. Universal hierarchical structure of quasiperiodic eigenfunctions. Ann. of Math. (2) 187(3) (2018), 721776.
[28]Jitomirskaya, S. and Liu, W.. Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase. Preprint, 2018, arXiv:1802.00781.
[29]Jitomirskaya, S. and Marx, C. A.. Analytic quasi-periodic Schrödinger operators and rational frequency approximants. Geom. Funct. Anal. 22(5) (2012), 14071443.
[30]Jitomirskaya, S. and Simon, B.. Operators with singular continuous spectrum. III. Almost periodic Schrödinger operators. Commun. Math. Phys. 165(1) (1994), 201205.
[31]Jitomirskaya, S. and Zhang, S.. Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators. Preprint, 2015, arXiv:1510.07086.
[32]Johnson, R. and Moser, J.. The rotation number for almost periodic potentials. Commun. Math. Phys. 84(3) (1982), 403438.
[33]Krasovsky, I.. Central spectral gaps of the almost Mathieu operator. Commun. Math. Phys. 351(1) (2017), 419439.
[34]Last, Y.. Spectral theory of Sturm–Liouville operators on infinite intervals: a review of recent developments. Sturm–Liouville Theory. Birkhäuser, Basel, 2005, pp. 99120.
[35]Last, Y. and Shamis, M.. Zero Hausdorff dimension spectrum for the almost Mathieu operator. Commun. Math. Phys. 348(3) (2016), 729750.
[36]Liu, W. and Shi, Y.. Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies. J. Spectr. Theory, to appear.
[37]Liu, W. and Yuan, X.. Anderson localization for the almost Mathieu operator in the exponential regime. J. Spectr. Theory 5(1) (2015), 89112.
[38]Liu, W. and Yuan, X.. Anderson localization for the completely resonant phases. J. Funct. Anal. 268(3) (2015), 732747.
[39]Liu, W. and Yuan, X.. Spectral gaps of almost Mathieu operators in the exponential regime. J. Fractal Geom. 2(1) (2015), 151.
[40]Marx, C. A. and Jitomirskaya, S.. Dynamics and spectral theory of quasi-periodic Schrödinger-type operators. Ergod. Th. & Dynam. Sys. 37(8) (2017), 23532393.
[41]Moser, J. and Pöschel, J.. An extension of a result by Dinaburg and Sinai on quasi-periodic potentials. Comment. Math. Helv. 59(1) (1984), 3985.
[42]Puig, J.. Cantor spectrum for the almost Mathieu operator. Commun. Math. Phys. 244(2) (2004), 297309.
[43]Puig, J.. A nonperturbative Eliasson’s reducibility theorem. Nonlinearity 19(2) (2006), 355376.
[44]Simon, B.. Almost periodic Schrödinger operators: a review. Adv. Appl. Math. 3(4) (1982), 463490.
[45]Yang, F.. Spectral transition line for the extended Harper’s model in the positive Lyapunov exponent regime. J. Funct. Anal. 275(3) (2018), 712734.


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Almost Mathieu operators with completely resonant phases

  • WENCAI LIU (a1)


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