Hostname: page-component-76c49bb84f-lvrpm Total loading time: 0 Render date: 2025-07-10T18:49:36.193Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics and spectral theory of quasi-periodic Schrödinger-type operators

Published online by Cambridge University Press:  04 July 2016

C. A. MARX  and
S. JITOMIRSKAYA
C. A. MARX
Affiliation:
Department of Mathematics, Oberlin College, Oberlin, OH 44074, USA email cmarx@oberlin.edu
S. JITOMIRSKAYA
Affiliation:
Department of Mathematics, University of California, Irvine, CA 92717, USA email szhitomi@uci.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We survey the theory of quasi-periodic Schrödinger-type operators, focusing on the advances made since the early 2000s by adopting a dynamical systems point of view.

Information

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 8 , December 2017 , pp. 2353 - 2393
Copyright
© Cambridge University Press, 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aubry, S. and André, G.. Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Isr. Phys. Soc. 3 (1980), 133–164.Google Scholar
Avila, A.. Absolutely continuous spectrum for the almost Mathieu operator with subcritical coupling. Preprint, 2008, arXiv:0810.2965v1.Google Scholar
Avila, A.. Density of positive Lyapunov exponents for SL (2, ℝ) cocycles. J. Amer. Math. Soc. 24 (2011), 999–1014.Google Scholar
Avila, A.. Almost reducibility and absolute continuity I. Preprint, 2011, arXiv:1006.0704v1.Google Scholar
Avila, A.. Global theory of one-frequency Schrödinger operators. Acta Math. 215 (2015), 1–54.Google Scholar
Avila, A.. On the Kotani–Last and Schrödinger conjectures. J. Amer. Math. Soc. 28 (2015), 579–616.Google Scholar
Avila, A.. On point spectrum with critical coupling. Preprint, 2008, available at http://www.impa.br/∼avila/.Google Scholar
Avila, A.. Almost reducibility and absolute continuity II, in preparation.Google Scholar
Avila, A. and Bochi, J.. A formula with some applications to the theory of Lyapunov exponent. Israel J. Math. 131 (2002), 125–137.Google Scholar
Avila, A. and Damanik, D.. Generic singular spectrum for ergodic Schrödinger operators. Duke Math. J. 130 (2005), 393–400.Google Scholar
Avila, A. and Damanik, D.. Absolute continuity of the integrated density of states for the almost Mathieu operator with non-critical coupling. Invent. Math. 172 (2008), 439–453.Google Scholar
Avila, A., Fayad, B. and Krikorian, R.. A KAM scheme for SL(2, ℝ) cocycles with Liouvillian frequencies. Geom. Funct. Anal. 21 (2011), 1001–1019.Google Scholar
Avila, A. and Jitomirskaya, S.. The ten Martini problem. Ann. of Math. (2) 170 (2009), 303–342.Google Scholar
Avila, A. and Jitomirskaya, S.. Almost localization and almost reducibility. J. Eur. Math. Soc. 12 (2010), 93–131.Google Scholar
Avila, A., Jitomirskaya, S. and Marx, C. A.. Spectral theory of extended Harper’s model and a question by Erdős and Szekeres. Preprint, 2016, arXiv:1602.05111.Google Scholar
Avila, A., Jitomirskaya, S. and Sadel, C.. Complex one-frequency cocycles. J. Eur. Math. Soc. 16(9) (2014), 1915–1935.CrossRefGoogle Scholar
Avila, A., Jitomirskaya, S. and Zhou, Q.. Second phase transition line. Preprint.Google Scholar
Avila, A. and Krikorian, R.. Reducibility and non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. of Math. (2) 164 (2006), 911–940.CrossRefGoogle Scholar
Avila, A. and Krikorian, R.. Reducibility and non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. An extended preprint of [ 18 ], 2006, available at http://w3.impa.br/∼avila/.Google Scholar
Avila, A. and Krikorian, R.. Monotonic cocycles. Invent. Math. 202(1) (2015), 271–331.Google Scholar
Avila, A., You, J. and Zhou, Q.. Sharp phase transitions for the almost Mathieu operator. Preprint, 2015,arXiv:1512.03124.Google Scholar
Avila, A., You, J. and Zhou, Q.. Dry ten Martini problem in the non-critical case, in preparation.Google Scholar
Avron, J., v. Mouche, P. H. M. and Simon, B.. On the measure of the spectrum for the almost Mathieu operator. Comm. Math. Phys. 132 (1990), 103–118.Google Scholar
Avron, J., Osadchy, D. and Seiler, R.. A topological look at the quantum Hall effect. Phys. Today 56(8) (2003), 38–42.Google Scholar
Avron, J. and Simon, B.. Almost periodic Schrödinger operators. II. The integrated density of states. Duke Math. J. 50 (1983), 369–391.Google Scholar
Bellissard, J.. The Gap Labelling Theorem: The Case of Automatic Sequences in Quantum and Non-commutative Analysis (Kyoto, 1992) (Mathematical Physics Studies, 16) . Kluwer Academic, Dordrecht, 1993, pp. 179–181.Google Scholar
Berezanskii, Y. M.. Expansions in Eigenfunctions of Self-Adjoint Operators. American Mathematical Society, Providence, RI, 1968.Google Scholar
Berry, M.. Incommensurability in an exactly-soluble quantal and classical model for a kicked rotator. Physica D 10 (1984), 369–378.Google Scholar
Binder, I. and Voda, M.. An estimate on the number of eigenvalues of a quasiperiodic Jacobi matrix of size n contained in an interval of size n -C . J. Spectr. Theory 3 (2013), 1–45.Google Scholar
Binder, I. and Voda, M.. On optimal separation of eigenvalues for a quasiperiodic Jacobi matrix. Comm. Math. Phys. 325 (2014), 1063–1106.Google Scholar
Bjerklov, K.. Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations. Ergod. Th. & Dynam. Sys. 25 (2005), 1015–1045.CrossRefGoogle Scholar
Bochi, J.. Genericity of zero Lyapunov exponents. Ergod. Th. & Dynam. Sys. 22(6) (2002), 1667–1696.Google Scholar
Bochi, J. and Viana, M.. The Lyapunov exponents of generic volume-preserving and symplectic maps. Ann. of Math. (2) 161 (2005), 1423–1485.Google Scholar
Boshernitzan, M. and Damanik, D.. Generic continuous spectrum for ergodic Schrödinger operators. Comm. Math. Phys. 283 (2008), 647–662.Google Scholar
Bourgain, J.. Green’s Function Estimates for Lattice Schrödinger Operators and Applications. Princeton University Press, Princeton, NJ, 2005.Google Scholar
Bourgain, J.. Positivity and continuity of the Lyapunov exponent for shifts on T d with arbitrary frequency vector and real analytic potential. J. Anal. Math. 96 (2005), 313–355.Google Scholar
Bourgain, J. and Goldstein, M.. On nonperturbative localization with quasi-periodic potential. Ann. of Math. (2) 152 (2000), 835–839.Google Scholar
Bourgain, J. and Jitomirskaya, S.. Anderson Localization for the Band Model (Lecture Notes in Mathematics, 1745) . Springer, Berlin, 2000, pp. 67–79.Google Scholar
Bourgain, J. and Jitomirskaya, S.. Continuity of the Lyapunov exponent for quasi-periodic operators with analytic potential. J. Stat. Phys. 108(5–6) (2002), 1203–1218.Google Scholar
Bourgain, J. and Jitomirskaya, S.. Absolutely continuous spectrum for 1D quasiperiodic operators. Invent. Math. 148 (2002), 453–463.CrossRefGoogle Scholar
Bourgain, J. and Klein, A.. Bounds on the density of states for Schrödinger operators. Invent. Math. 194 (2013), 41–72.Google Scholar
Chan, J., Goldstein, M. and Schlag, W.. On non-perturbative Anderson localization for ${\mathcal{C}}^{\unicode[STIX]{x1D6FC}}$ potentials generated by shifts and skew-shifts. Preprint, 2006, arXiv:math/06077302v2 [math.DS].Google Scholar
Choi, M. D., Eliott, G. A. and Yui, N.. Gauss polynomials and the rotation algebra. Invent. Math. 99 (1990), 225–246.CrossRefGoogle Scholar
Chulaevsky, V. and Delyon, F.. Purely absolutely continuous spectrum for almost Mathieu operators. J. Stat. Phys. 55 (1989), 1279–1284.CrossRefGoogle Scholar
Craig, W.. Point spectrum for discrete almost periodic Schrödinger operators. Comm. Math. Phys. 88 (1983), 113–131.Google Scholar
Craig, W. and Simon, B.. Subharmonicity of the Lyapunov index. Duke Math. J. 50 (1983), 551–560.Google Scholar
Cycon, H., Froese, R., Kirsch, W. and Simon, B.. Schrödinger Operators with Application to Quantum Mechanics and Global Geometry (Texts and Monographs in Physics) . Springer, Berlin, 1987.Google Scholar
Damanik, D.. Lyapunov exponent and spectral analysis of ergodic Schrödinger operators: a survey of Kotani theory and its applications. Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday (Proceedings of Symposia in Pure Mathematics, 76, Part 2) . American Mathematical Society, Providence, RI, 2007.Google Scholar
Damanik, D.. Schrödinger operators with dynamically defined potentials: a survey. Ergod. Th. & Dynam. Sys., to appear.Google Scholar
Damanik, D., Goldstein, M., Schlag, W. and Voda, M.. Homogeneity of the spectrum for quasi-perioidic Schrödinger operators. Preprint, 2015, arXiv:1505.04904.Google Scholar
Damanik, D. and Killip, R.. Ergodic potentials with a discontinuous sampling function are non-deterministic. Math. Res. Lett. 12 (2005), 187–193.Google Scholar
Damanik, D., Killip, R. and Simon, B.. Perturbations of orthogonal polynomials with periodic recursion coefficients. Ann. of Math. (2) 171 (2010), 1931–2010.Google Scholar
Damanik, D. and Tcheremchantzev, S.. Upper bounds in quantum dynamics. J. Adv. Math. Stud. 20 (2007), 799–827.Google Scholar
Deift, P. and Simon, B.. Almost periodic Schrödinger operators, III. The absolutely continuous spectrum in one dimension. Comm. Math. Phys. 90 (1983), 389–411.Google Scholar
Delyon, F.. Absence of localization for the almost Mathieu equation. J. Phys. A: Math. Gen. 20 (1987), L21–L23.Google Scholar
Delyon, F. and Souillard, B.. The rotation number for finite difference operators and its properties. Comm. Math. Phys. 89 (1983), 415–426.Google Scholar
Delyon, F. and Souillard, B.. Remark on the continuity of the density of states of ergodic finite difference operators. Comm. Math. Phys. 94 (1984), 289–291.Google Scholar
Dinaburg, E. I. and Sinai, Ya. G.. The one-dimensional Schrödinger equation with a quasi-periodic potential. Funct. Anal. Appl. 9 (1975), 279–289.Google Scholar
Disertori, M., Kirsch, W., Klein, A., Klopp, F. and Rivasseau, V.. Random Schrödinger Operators (Panoramas et Syntheses, 25) . Société de Mathématique de France, Paris, 2008.Google Scholar
Dombrowsky, J.. Quasitriangular matrices. Proc. Amer. Math. Soc. 69 (1978), 95–96.Google Scholar
Duarte, P. and Klein, S.. Positive Lyapunov exponents for higher dimensional quasiperiodic cocycles. Comm. Math. Phys. 332 (2014), 189–219.Google Scholar
Duarte, P. and Klein, S.. Continuity of the Lyapunov exponents for quasi-periodic cocycles. Comm. Math. Phys. 332 (2014), 1113–1166.Google Scholar
Duren, P. L.. Theory of H p Spaces (Pure and Applied Mathematics, 38) . Academic Press, New York, 1970.Google Scholar
Eliasson, L. H.. Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Comm. Math. Phys. 146 (1982), 447–482.Google Scholar
Eliasson, L. H.. Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum. Acta Math. 179 (1997), 153–196.Google Scholar
Figotin, A. L. and Pastur, L. A.. An exactly solvable model of a multidimensional incommensurate structure. Comm. Math. Phys. 95(4) (1984), 401–425.CrossRefGoogle Scholar
Fröhlich, J., Spencer, T. and Wittwer, P.. Localization for a class of one dimensional quasi-periodic Schrödinger operators. Comm. Math. Phys. 132 (1990), 5–25.CrossRefGoogle Scholar
Furman, A.. On the multiplicative ergodic theorem for uniquely ergodic systems. Ann. Inst. Henri Poincaré Probab. Stat. 33(6) (1997), 797–815.CrossRefGoogle Scholar
Ganeshan, S., Kechedzhi, K. and Das Sarma, S.. Critical integer quantum hall topology and the integrable Maryland model as a topological quantum critical point. Phys. Rev. B 90 (2014), 041405.Google Scholar
Goldsheid, I. Ya. and Sorets, E.. Lyapunov exponents of the Schrödinger equation with quasi-periodic potential on a strip. Comm. Math. Phys. 145 (2012), 507–513.Google Scholar
Goldstein, M. and Schlag, W.. Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. of Math. (2) 154 (2001), 155–203.CrossRefGoogle Scholar
Goldstein, M. and Schlag, W.. Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues. Geom. Funct. Anal. 18 (2008), 765–869.CrossRefGoogle Scholar
Goldstein, M. and Schlag, W.. On resonances and the formation of gaps in the spectrum of quasi-periodic Schroedinger equations. Ann. of Math. (2) 173 (2011), 337–475.CrossRefGoogle Scholar
Gordon, A.. The point spectrum of the one-dimensional Schrödinger operator. Uspekhi Mat. Nauk 31 (1976), 257–258.Google Scholar
Gordon, A. Y., Jitomirskaya, S., Last, Y. and Simon, B.. Duality and singular continuous spectrum in the almost Mathieu equation. Acta Math. 178 (1997), 169–183.Google Scholar
Grempel, D., Fishman, S. and Prange, R.. Localization in an incommensurate potential: an exactly solvable model. Phys. Rev. Lett. 49(11) (1982), 833–836.CrossRefGoogle Scholar
Hadj Amor, S.. Hölder continuity of the rotation number for quasi-periodic co-cycles in SL (2, ℝ). Comm. Math. Phys. 287 (2009), 565–588.Google Scholar
Han, R.. Dry ten martini problem for the non-self-dual extended Harper’s model. Preprint.Google Scholar
Han, R. and Jitomirskaya, S.. The total bandwidth of the extended Harper’s model, in preparation.Google Scholar
Han, R. and Jitomirskaya, S.. In preparation.Google Scholar
Haro, A. and Puig, J.. A Thouless formula and Aubry duality for long-range Schrödinger skew-products. Nonlinearity 26 (2013), 1163–1187.Google Scholar
Harper, P. G.. Single band motion of conducting electrons in a uniform magnetic field. Proc. Phys. Soc. A 68 (1955), 874–878.CrossRefGoogle Scholar
Herman, M.. Une methode pour minorer les exposants des Lyapunov et quelques examples montrant le charactère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58 (1983), 453–562.Google Scholar
Hofstadter, D. R.. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14 (1976), 2239–2249.Google Scholar
Jitomirskaya, S.. Almost everything about the almost Mathieu operator. II. Proc. XIth Int. Congr. Mathematical Physics. Ed. Iagolnitzer, D.. International Press, Cambridge, MA, 1995, pp. 373–382.Google Scholar
Jitomirskaya, S. Ya.. Metal–insulator transition for the almost Mathieu operator. Ann. of Math. (2) 150 (1999), 1159–1175.Google Scholar
Jitomirskaya, S.. Ergodic Schrödinger operators (on one foot). Proc. Sympos. Pure Math. 76 (2007), 613–647.CrossRefGoogle Scholar
Jitomirskaya, S. and Kachkovskiy, I.. Anderson localization for discrete 1D quasiperiodic operators with piecewise monotonic sampling functions. Preprint, 2015.Google Scholar
Jitomirskaya, S. and Kachkovsky, I.. $L^{2}$ -reducibility and localization for quasiperiodic operators. Math. Res. Lett., to appear.Google Scholar
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 (2005), 103–124.CrossRefGoogle Scholar
Jitomirskaya, S. and Krasovsky, I.. Continuity of the measure of the spectrum for discrete quasiperiodic operators. Math. Res. Lett. 9 (2002), 413–421.Google Scholar
Jitomirskaya, S. and Krüger, H.. Exponential dynamical localization for the almost Mathieu operator. Comm. Math. Phys. 322 (2013), 877–882.Google Scholar
Jitomirskaya, S., Krüger, H. and Liu, W.. In preparation.Google Scholar
Jitomirskaya, S. and Last, Y.. Power-law subordinacy and singular spectra, II. Line operators. Comm. Math. Phys. 211 (2000), 643–658.Google Scholar
Jitomirskaya, S. and Liu, W.. Arithmetic spectral transitions for the Maryland model. Preprint, 2014.Google Scholar
Jitomirskaya, S. and Liu, W.. Asymptotics of quasiperiodic eigenfunctions. Preprint.Google Scholar
Jitomirskaya, S. and Liu, W.. Asymptotics of quasiperiodic eigenfunctions, II. Preprint.Google Scholar
Jitomirskaya, S., Liu, W. and Tcheremchancev, S.. In preparation.Google Scholar
Jitomirskaya, S. and Marx, C. A.. Continuity of the Lyapunov exponent for analytic quasi-periodic cocycles with singularities. J. Fixed Point Theory Appl. 10 (2011), 129–146.Google Scholar
Jitomirskaya, S. and Marx, C. A.. Analytic quasi-periodic Schrödinger operators and rational frequency approximation. Geom. Funct. Anal. 22 (2012), 1407–1443.Google Scholar
Jitomirskaya, S. and Marx, C. A.. Analytic quasi-periodic cocycles with singularities and the Lyapunov exponent of extended Harper’s model. Comm. Math. Phys. 316 (2012), 237–267.Google Scholar
Jitomirskaya, S. and Marx, C. A.. Erratum to: analytic quasi-periodic cocycles with singularities and the Lyapunov exponent of extended Harper’s model. Comm. Math. Phys. 317 (2013), 269–271.Google Scholar
Jitomirskaya, S. and Mavi, R.. Continuity of the measure of the spectrum for quasiperiodic Schrödinger operators with rough potentials. Comm. Math. Phys. 325 (2014), 585–601.Google Scholar
Jitomirskaya, S. and Mavi, R.. Dynamical bounds for quasiperiodic Schrödinger operators with rough potentials. Preprint, 2015, arXiv:1412.0309.Google Scholar
Jitomirskaya, S. and Simon, B.. Operators with singular continuous spectrum: III. Almost periodic Schrödinger operators. Comm. Math. Phys. 165 (1994), 201–205.Google Scholar
Jitomirskaya, S. and Zhang, S.. Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators. Preprint, 2015, arXiv:1510.07086 [math.SP].Google Scholar
Jitomirskaya, S. and Zhou, Q.. In preparation.Google Scholar
Johnson, R. A.. Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients. J. Differential Equations 61 (1986), 54–78.Google Scholar
Johnson, R. and Moser, J.. The rotation number for almost periodic potentials. Comm. Math. Phys. 84 (1982), 403–438.Google Scholar
Kirsch, W.. An invitation to random Schrödinger operators. Preprint, 2007, arXiv:0709.3707 [math-ph].Google Scholar
Kirsch, W. and Martinelli, F.. On the ergodic properties of the spectrum of general random operators. J. Reine Angew. Math. 334 (1982), 141–156.Google Scholar
Klein, S.. Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function. J. Funct. Anal. 218 (2005), 255–292.Google Scholar
Klein, S.. Localization for quasi-periodic Schrödinger operators with multivariable Gevrey potential functions. J. Spectr. Theory 4(3) (2014), 431–484.Google Scholar
Klein, A., Koines, A. and Seifert, M.. Generalized eigenfunctions for waves in inhomogeneous media. J. Funct. Anal. 190 (2002), 255–291.Google Scholar
Koslover, D. A.. Jacobi operators with singular continuous spectrum. Lett. Math. Phys. 71 (2005), 123–134.Google Scholar
Kotani, S.. Lyapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators. Proc. Taniguchi Symp. on Stochastic Analysis (Katata, 1982).Google Scholar
Kotani, S.. Generalized Floquet theory for stationary Schrödinger operators in one dimension. Chaos Solitons Fractals 8 (1997), 1817–1854.Google Scholar
Kotani, S. and Simon, B.. Stochastic Schrödinger operators and Jacobi matrices on the strip. Comm. Math. Phys. 119 (1988), 403–429.Google Scholar
Krasovsky, I.. Bloch electron in a magnetic field and the Ising model. Phys. Rev. Lett. 85 (2000), 4920–4923.Google Scholar
Krüger, H. and Zheng, G.. Optimality of log Hölder continuity of the integrated density of states. Math. Nachr. 284 (2011), 1919–1923.Google Scholar
Kunz, H. and Souillard, B.. Sur le spectre des opérateurs aux différences finies aléatoires. Comm. Math. Phys. 78 (1980), 201–246.Google Scholar
Last, Y.. A relation between absolutely continuous spectrum of ergodic Jacobi matrices and the spectra of periodic approximants. Comm. Math. Phys. 151 (1993), 183–192.Google Scholar
Last, Y.. Zero measure spectrum for the almost Mathieu operator. Comm. Math. Phys. 164 (1994), 421–432.Google Scholar
Last, Y.. Spectral theory of Sturm–Liouville operators on infinite intervals: a review of recent developments. Sturm–Liouville Theory. Birkhäuser, Basel, 2005, pp. 99–120.Google Scholar
Last, Y. and Shamis, M.. Zero Hausdorff dimension spectrum for the almost Mathieu operator. Preprint, 2015, arXiv:1510.07651.Google Scholar
Last, Y. and Simon, B.. Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators. Invent. Math. 135 (1999), 329–367.Google Scholar
Liu, W. and Yuan, X.. Anderson localization for the almost Mathieu operator in exponential regime.J. Spectr. Theory, to appear.Google Scholar
Luttinger, J. M.. The effect of a magnetic field on electrons in a periodic potential. Phys. Rev. 84 (1951), 814–817.Google Scholar
Mandel’shtam, V. A. and Zhitomirskaya, S. Ya.. 1D-quasiperiodic operators. Latent symmetries. Comm. Math. Phys. 139 (1991), 589–604.Google Scholar
Marx, C. A.. Quasi-periodic Jacobi-cocycles: dynamics, continuity, and applications to extended Harper’s model. PhD Thesis, Irvine, CA, 2012.Google Scholar
Marx, C. A.. Dominated splittings and the spectrum for quasi periodic Jacobi operators. Nonlinearity 27 (2014), 3059–3072.Google Scholar
Marx, C. A., Shou, L. and Wellens, J.. A sufficient criterion of subcritical behavior in quasi-periodic Jacobi operators with trigonometric sampling functions. J. Spectr. Theory, to appear.Google Scholar
Moser, J. and Pöschel, J.. An extension of a result by Dinaburg and Sinai on quasi-periodic potentials. Comment. Math. Helv. 59 (1984), 39–85.Google Scholar
Oseledets, V. I.. A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems. Tr. Mosk. Mat. Obs. 19 (1968), 179–210. Engl. transl. Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
Pastur, L.. Spectral properties of disordered systems in one-body approximation. Comm. Math. Phys. 75 (1980), 179–196.Google Scholar
Pastur, L. and Figotin, A.. Spectra of Random and Almost-periodic Operators (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 297) . Springer, Berlin, 1992.Google Scholar
Peierls, R.. On the theory of diamagnetism of conduction electrons. Z. Phys. 80 (1933), 763–791.Google Scholar
Pöschel, J.. Examples of discrete Schrödinger operators with pure point spectrum. Comm. Math. Phys. 88 (1983), 447–463.Google Scholar
Puig, J.. Cantor spectrum for the almost Mathieu operator. Comm. Math. Phys. 244 (2004), 297–309.Google Scholar
Puig, J.. A nonperturbative Eliasson’s reducibility theorem. Nonlinearity 19 (2006), 355–376.Google Scholar
Rauh, A.. Degeneracy of Landau levels in crystals. Phys. Status Solidi B 65 (1974), K131–K135.CrossRefGoogle Scholar
Ruelle, D.. Ergodic theory of differentiable dynamical systems. Publ. Math. Hautes Études Sci. 50 (1979), 27–58.Google Scholar
Ruelle, D.. Analyticity properties of the characteristic exponents of random matrix products. Adv. Math. 32 (1979), 68–80.Google Scholar
Schlag, W.. Regularity and convergence rate for the Lyapunov exponents of linear cocycles. J. Mod. Dyn. 7 (2013), 619–637.Google Scholar
Sch’nol, I.. On the behavior of the Schrödinger equation. Mat. Sb. 42 (1957), 273–286.Google Scholar
Shamis, M.. Spectral analysis of Jacobi operators. PhD Thesis, Hebrew University of Jerusalem, 2010.Google Scholar
Shamis, M. and Spencer, Th.. Bounds on the Lyapunov exponent via crude estimates on the density of states. Comm. Math. Phys. 338(2) (2015), 705–720.Google Scholar
Simon, B.. Almost periodic Schrö dinger operators: a review. Adv. Appl. Math. 3 (1982), 463–490.Google Scholar
Simon, B.. Schrödinger semigroups. Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447–526.Google Scholar
Simon, B.. Kotani theory for one dimensional stochastic Jacobi matrices. Comm. Math. Phys. 89 (1983), 227–234.Google Scholar
Simon, B.. Fifteen problems in mathematical physics. Perspectives in Mathematics. Oberwolfach Anniversary Volume. Birkhauser, Basel, 1984, pp. 423–454.Google Scholar
Simon, B.. Almost periodic Schrödinger operators. IV. The Maryland model. Ann. Phys. 159(1) (1985), 157–183.Google Scholar
Simon, B.. Schrödinger operators in the twenty-first century. Math. Phys. 2000 (2000), 283–288.Google Scholar
Simon, B.. Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory (American Mathematical Society Colloquium Publications, 54) . American Mathematical Society, Providence, RI, 2005.Google Scholar
Simon, B.. Equilibrium measures and capacities in spectral theory. Inverse Probl. Imaging 1 (2007), 713–772.Google Scholar
Sinai, Ya.. Anderson localization for one-dimensional difference Schrödinger operator with quasi-periodic potential. J. Stat. Phys. 46 (1987), 861–909.Google Scholar
Sorets, E. and Spencer, T.. Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials. Comm. Math. Phys. 142(3) (1991), 543–566.CrossRefGoogle Scholar
Spencer, T.. Ergodic Schrödinger operators. Analysis, et cetera. Academic Press, Boston, MA, 1990, pp. 623–637.Google Scholar
Tao, K.. Continuity of Lyapunov exponent for analytic quasi-periodic cocycles on higher-dimensional torus. Front. Math. China 7(3) (2012), 521–542.Google Scholar
Tao, K.. Hölder continuity of Lyapunov exponent for quasi-periodic Jacobi operators. Bull. Soc. Math. France 142(4) (2014), 635–671.Google Scholar
Tao, K. and Voda, M.. Hölder continuity of the integrated density of states for quasi-periodic Jacobi operators. Preprint, 2015, arXiv:1501.01028 [math.SP].Google Scholar
Teschl, G.. Jacobi Operators and Completely Integrable Nonlinear Lattices (Mathematical Surveys and Monographs, 72) . American Mathematical Society, Providence, RI, 2000.Google Scholar
Thouless, D.. A relation between the density of states and range of localization for one-dimensional random system. J. Phys. C: Solid State Phys. 5 (1972), 77–81.Google Scholar
Thouless, D. J.. Bandwidth for a quasiperiodic tight binding model. Phys. Rev. B 28 (1983), 4272–4276.Google Scholar
Thouless, D. J., Kohmoto, M., Nightingale, M. P. and den Nijs, M.. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49 (1982), 405–408.Google Scholar
Wang, Y. and You, J.. Examples of discontinuity of Lyapunov exponent in smooth quasi-periodic cocycles. Duke Math. J. 162(13) 2363–2412.Google Scholar
Wang, Y. and You, J.. Quasi-periodic Schrödinger cocycles with positive Lyapunov exponent are not open in the smooth topology. Preprint, 2015, arXiv:1501.05380.Google Scholar
Wang, Y. and Zhang, Z.. Uniform positivity and continuity of Lyapunov exponents for a class of C 2 quasiperiodic Schrödinger cocycles. J. Funct. Anal. 268 (2015), 2525–2585.Google Scholar
Wang, Y. and Zhang, Z.. Cantor spectrum for a class of $C^{2}$ quasiperiodic Schrödinger operators. Int. Math. Res. Not. IMRN, to appear.Google Scholar
You, J. and Zhou, Q.. Embedding of analytic quasi-periodic cocycles into analytic quasi-periodic linear systems and its applications. Comm. Math Phys. 323 (2013), 975–1005.Google Scholar
Young, L.-S.. Lyapunov exponents for some quasi-periodic cocycles. Ergod. Th. & Dynam. Sys. 17 (1997), 483–504.Google Scholar
Zhang, Z.. Positive Lyapunov exponents for quasiperiodic Szegő cocycles. Nonlinearity 25 (2012), 1771–1797.Google Scholar
Zhang, Z.. Resolvent set of Schrödinger operators and uniform hyperbolicty. Preprint, 2013, arXiv:1305.4226v1.Google Scholar