Abstract
We obtain the asymptotics of the optimal global Hölder exponent of the integrated density of states of the Fibonacci Hamiltonian for large and small couplings.
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Avila A., Jitomirskaya S.: Almost localization and almost reducibility. J. Eur. Math. Soc. (JEMS) 12, 93–131 (2010)
Avron J., Simon B.: Almost periodic Schrödinger operators. II. The integrated density of states. Duke Math. J. 50, 369–391 (1983)
Baake, M., Roberts, J.: The dynamics of trace maps. In: Hamiltonian Mechanics (Toruń, 1993) NATO Adv. Sci. Inst. Ser. B Phys. 331, New York: Plenum, 1994, pp. 275–285
Bourgain J.: Hölder regularity of integrated density of states for the almost Mathieu operator in a perturbative regime. Lett. Math. Phys. 51, 83–118 (2000)
Bourgain, J.: Green’s function estimates for lattice Schrödinger operators and applications. Ann. Math. Stud. 158, Princeton, NJ: Princeton University Press, 2005
Bourgain J., Goldstein M., Schlag M.: Anderson localization for Schrödinger operators on \({\mathbb {Z}}\) with potentials given by the skew-shift. Commun. Math. Phys. 220, 583–621 (2001)
Campanino M., Klein A.: A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model. Commun. Math. Phys. 104, 227–241 (1986)
Cantat S.: Bers and Hénon, Painlevé and Schrödinger. Duke Math. J. 149, 411–460 (2009)
Casdagli M.: Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation. Commun. Math. Phys. 107, 295–318 (1986)
Craig W.: Pure point spectrum for discrete almost periodic Schrödinger operators. Commun. Math. Phys. 88, 113–131 (1983)
Craig W., Simon B.: Subharmonicity of the Lyaponov index. Duke Math. J. 50, 551–560 (1983)
Craig W., Simon B.: Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices. Commun. Math. Phys. 90, 207–218 (1983)
Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Texts and Monographs in Physics, Berlin: Springer-Verlag, 1987
Damanik D.: α-continuity properties of one-dimensional quasicrystals. Commun. Math. Phys. 192, 169–182 (1998)
Damanik, D.: Gordon-type arguments in the spectral theory of one-dimensional quasicrystals. In: Directions in Mathematical Quasicrystals, Eds. M. Baake, R. V. Moody, CRM Monograph Series 13, Providence, RI: Amer. Math. Soc., 2000, pp. 277–305
Damanik, D.: Strictly ergodic subshifts and associated operators. In:Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon’s 60th Birthday, Proc. Sympos. Pure Math. 76, Part 2, Providence, RI: Amer. Math. Soc., 2007, pp. 505–538.
Damanik, D.: Almost everything about the Fibonacci operator. In: New Trends in Mathematical Physics, Selected contributions of the XVth International Congress on Mathematical Physics, Berlin-Heidelberg-New York: Springer 2009, pp. 149–159
Damanik D., Embree M., Gorodetski A., Tcheremchantsev S.: The fractal dimension of the spectrum of the Fibonacci Hamiltonian. Commun. Math. Phys. 280, 499–516 (2008)
Damanik D., Gorodetski A.: Hyperbolicity of the trace map for the weakly coupled Fibonacci Hamiltonian. Nonlinearity 22, 123–143 (2009)
Damanik D., Gorodetski A.: The spectrum of the weakly coupled Fibonacci Hamiltonian. Electron. Res. Annc. Math. Sci. 16, 23–29 (2009)
Damanik D., Gorodetski A.: Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian. Commun. Math. Phys. 305, 221–277 (2011)
Damanik D., Gorodetski A.: The density of states measure of the weakly coupled Fibonacci Hamiltonian. Geom. Funct. Anal. 22, 976–989 (2012)
Damanik D., Killip R., Lenz D.: Uniform spectral properties of one-dimensional quasicrystals. III. α-continuity, Commun. Math. Phys. 212, 191–204 (2000)
Damanik D., Lenz D.: Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues. Commun. Math. Phys. 207, 687–696 (1999)
Damanik D., Tcheremchantsev S.: Power-law bounds on transfer matrices and quantum dynamics in one dimension. Commun. Math. Phys. 236, 513–534 (2003)
Damanik D., Tcheremchantsev S.: Upper bounds in quantum dynamics. J. Amer. Math. Soc. 20, 799–827 (2007)
Delyon F., Souillard B.: Remark on the continuity of the density of states of ergodic finite difference operators. Commun. Math. Phys. 94, 289–291 (1984)
Gan Z., Krüger H.: Optimality of log Hölder continuity of the integrated density of states. Math. Nachr. 284, 1919–1923 (2011)
Goldstein M., 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. 154, 155–203 (2001)
Goldstein M., Schlag W.: Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues. Geom. Funct. Anal. 18, 755–869 (2008)
Hadj Amor S.: Hölder continuity of the rotation number for quasi-periodic co-cycles in \({\rm{SL}(2,\mathbb {R})}\) . Commun. Math. Phys. 287, 565–588 (2009)
Hof A.: Some remarks on discrete aperiodic Schrödinger operators. J.Stat.Phys. 72, 1353–1374 (1993)
Iochum B., Testard D.: Power law growth for the resistance in the Fibonacci model. J. Stat. Phys. 65, 715–723 (1991)
Jitomirskaya S., Last Y.: Power law subordinacy and singular spectra. II. Line operators. Commun. Math. Phys. 211, 643–658 (2000)
Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press, 1995
Killip R., Kiselev A., Last Y.: Dynamical upper bounds on wavepacket spreading. Amer. J. Math. 125, 1165–1198 (2003)
Kohmoto M., Kadanoff L.P., Tang C.: Localization problem in one dimension: Mapping and escape. Phys. Rev. Lett. 50, 1870–1872 (1983)
Kotani S.: Jacobi matrices with random potentials taking finitely many values. Rev. Math. Phys. 1, 129–133 (1989)
Krüger, H., Simon, B.: Private communication and in preparation
Le Page, E.: State distribution of a random Schrödinger operator. Empirical distribution of the eigenvalues of a Jacobi matrix. In: Probability measures on groups, VII (Oberwolfach, 1983), Lecture Notes in Math. 1064, Berlin: Springer-Verlag, 1984, pp. 309–367
Ostlund S., Pandit R., Rand D., Schellnhuber H.J., Siggia E.D.: One-dimensional Schrödinger equation with an almost periodic potential. Phys. Rev. Lett. 50, 1873–1877 (1983)
Palis, J., Viana, M.: On the continuity of Hausdorff dimension and limit capacity for horseshoes. Lecture Notes in Math. 1331, Berlin: Springer, 1988
Raymond, L.: A constructive gap labelling for the discrete Schrödinger operator on a quasiperiodic chain. Preprint 1997
Roberts J.: Escaping orbits in trace maps. Phys. A 228, 295–325 (1996)
Schlag W.: On the integrated density of states for Schrödinger operators on \({\mathbb {Z}^2}\) with quasi periodic potential. Commun. Math. Phys. 223, 47–65 (2001)
Simon, B.: Regularity of the density of states for stochastic Jacobi matrices: a review. In: Random media (Minneapolis, Minn., 1985). IMA Vol. Math. Appl. 7, New York: Springer, 1987, pp. 245–266
Simon B., Taylor M.: Harmonic analysis on \({{\rm SL}(2, \mathbb {R})}\) and smoothness of the density of states in the one-dimensional Anderson model. Commun. Math. Phys. 101, 1–19 (1985)
Sütő A.: The spectrum of a quasiperiodic Schrödinger operator. Commun. Math. Phys. 111, 409–415 (1987)
Sütő A.: Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian. J. Stat. Phys. 56, 525–531 (1989)
Sütő, A.: Schrödinger difference equation with deterministic ergodic potentials. In: Beyond Quasicrystals (Les Houches, 1994), Eds. F. Axel, D. Gratias, Berlin: Springer, 1995, pp. 481–549
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Communicated by B. Simon
D. D. was supported in part by a Simons Fellowship and NSF grants DMS–0800100 and DMS–1067988.
A. G. was supported in part by NSF grants DMS–0901627 and IIS-1018433.
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Damanik, D., Gorodetski, A. Hölder Continuity of the Integrated Density of States for the Fibonacci Hamiltonian. Commun. Math. Phys. 323, 497–515 (2013). https://doi.org/10.1007/s00220-013-1753-7
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DOI: https://doi.org/10.1007/s00220-013-1753-7