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Mixed spectral types for the one-frequency discrete quasi-periodic Schrödinger operator


Author: Shiwen Zhang
Journal: Proc. Amer. Math. Soc. 144 (2016), 2603-2609
MSC (2010): Primary 37A30, 47B36, 82B44
DOI: https://doi.org/10.1090/proc/12929
Published electronically: October 19, 2015
MathSciNet review: 3477077
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a family of one-frequency discrete analytic quasi-periodic Schrödinger operators. We show that this family provides an example of coexistence of absolutely continuous and point spectrum for some parameters as well as coexistence of absolutely continuous and singular continuous spectrum for some other parameters.


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  • [A1] A. Avila, Almost reducibility and absolute continuity I, preprint.
  • [A2] A. Avila, Almost reducibility and absolute continuity II (in preparation).
  • [A3] A. Avila, Global theory of one-frequency Schrodinger operators II, preprint.
  • [Bo] J. Bourgain, On the spectrum of lattice Schrödinger operators with deterministic potential. II, J. Anal. Math. 88 (2002), 221-254, Dedicated to the memory of Tom Wolff. MR 1984594 (2004e:47046), https://doi.org/10.1007/BF02786578
  • [Bo1] J. Bourgain, Green's function estimates for lattice Schrödinger operators and applications, Annals of Mathematics Studies, vol. 158, Princeton University Press, Princeton, NJ, 2005. MR 2100420 (2005j:35184)
  • [BG] J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math. (2) 152 (2000), no. 3, 835-879. MR 1815703 (2002h:39028), https://doi.org/10.2307/2661356
  • [BJ] J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Statist. Phys. 108 (2002), no. 5-6, 1203-1218, Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. MR 1933451 (2004c:47073), https://doi.org/10.1023/A:1019751801035
  • [Bjer] K. Bjerklöv, Explicit examples of arbitrarily large analytic ergodic potentials with zero Lyapunov exponent, Geom. Funct. Anal. 16 (2006), no. 6, 1183-1200. MR 2276537 (2008b:47069), https://doi.org/10.1007/s00039-006-0581-8
  • [BK] K. Bjerklöv and R. Krikorian, Coexistence of ac and pp spectrum for quasiperiodic 1D Schrödinger operators (in preparation).
  • [CFKS] H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Springer Study Edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. MR 883643 (88g:35003)
  • [E] L. H. Eliasson, Floquet solutions for the $ 1$-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys. 146 (1992), no. 3, 447-482. MR 1167299 (93d:34141)
  • [FK] Alexander Fedotov and Frédéric Klopp, Coexistence of different spectral types for almost periodic Schrödinger equations in dimension one, Mathematical results in quantum mechanics (Prague, 1998) Oper. Theory Adv. Appl., vol. 108, Birkhäuser, Basel, 1999, pp. 243-251. MR 1708804 (2000g:34138)
  • [G1] A. Ja. Gordon, The point spectrum of the one-dimensional Schrödinger operator, Uspehi Mat. Nauk 31 (1976), no. 4(190), 257-258 (Russian). MR 0458247 (56 #16450)
  • [G2] Alexander Y. Gordon, A spectral alternative for continuous families of self-adjoint operators, J. Spectr. Theory 3 (2013), no. 2, 129-145. MR 3042762, https://doi.org/10.4171/JST/40
  • [GS1] Michael Goldstein and Wilhelm Schlag, 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), no. 1, 155-203. MR 1847592 (2002h:82055), https://doi.org/10.2307/3062114
  • [GS2] Michael Goldstein and Wilhelm Schlag, On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations, Ann. of Math. (2) 173 (2011), no. 1, 337-475. MR 2753606 (2012d:81100), https://doi.org/10.4007/annals.2011.173.1.9
  • [HY] Xuanji Hou and Jiangong You, Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems, Invent. Math. 190 (2012), no. 1, 209-260. MR 2969277, https://doi.org/10.1007/s00222-012-0379-2
  • [K] S. Kotani, Lyapunov indices determine absolutely continuous spectra of stationary random one-dimensional, Schrödinger operators, Proc. Kyoto Stoch. Conf., 1982.
  • [S] Barry Simon, Kotani theory for one-dimensional stochastic Jacobi matrices, Comm. Math. Phys. 89 (1983), no. 2, 227-234. MR 709464 (85d:60122)
  • [YZ] Jiangong You and Qi Zhou, Embedding of analytic quasi-periodic cocycles into analytic quasi-periodic linear systems and its applications, Comm. Math. Phys. 323 (2013), no. 3, 975-1005. MR 3106500, https://doi.org/10.1007/s00220-013-1800-4

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Additional Information

Shiwen Zhang
Affiliation: Department of Mathematics, University of California Irvine, Irvine, California 92617
Email: shiwez1@uci.edu

DOI: https://doi.org/10.1090/proc/12929
Keywords: Quasi-periodic Schr\"odinger operators, mixed spectral types, Lyapunov exponent, almost reducibility
Received by editor(s): April 24, 2015
Received by editor(s) in revised form: August 5, 2015
Published electronically: October 19, 2015
Additional Notes: This research was partially supported by NSF DMS-1401204
Communicated by: Michael Hitrik
Article copyright: © Copyright 2015 American Mathematical Society

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