Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Mixed spectral types for the one-frequency discrete quasi-periodic Schrödinger operator
HTML articles powered by AMS MathViewer

by Shiwen Zhang
Proc. Amer. Math. Soc. 144 (2016), 2603-2609
DOI: https://doi.org/10.1090/proc/12929
Published electronically: October 19, 2015
PDF | Request permission

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.
References
  • A. Avila, Almost reducibility and absolute continuity I, preprint.
  • A. Avila, Almost reducibility and absolute continuity II (in preparation).
  • A. Avila, Global theory of one-frequency Schrodinger operators II, preprint.
  • 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, DOI 10.1007/BF02786578
  • 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, DOI 10.1515/9781400837144
  • J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math. (2) 152 (2000), no. 3, 835–879. MR 1815703, DOI 10.2307/2661356
  • 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, DOI 10.1023/A:1019751801035
  • 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, DOI 10.1007/s00039-006-0581-8
  • K. Bjerklöv and R. Krikorian, Coexistence of ac and pp spectrum for quasiperiodic 1D Schrödinger operators (in preparation).
  • 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
  • L. H. Eliasson, Floquet solutions for the $1$-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys. 146 (1992), no. 3, 447–482. MR 1167299
  • 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
  • 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
  • Alexander Y. Gordon, A spectral alternative for continuous families of self-adjoint operators, J. Spectr. Theory 3 (2013), no. 2, 129–145. MR 3042762, DOI 10.4171/JST/40
  • 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, DOI 10.2307/3062114
  • 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, DOI 10.4007/annals.2011.173.1.9
  • 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, DOI 10.1007/s00222-012-0379-2
  • S. Kotani, Lyapunov indices determine absolutely continuous spectra of stationary random one-dimensional, Schrödinger operators, Proc. Kyoto Stoch. Conf., 1982.
  • Barry Simon, Kotani theory for one-dimensional stochastic Jacobi matrices, Comm. Math. Phys. 89 (1983), no. 2, 227–234. MR 709464
  • 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, DOI 10.1007/s00220-013-1800-4
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37A30, 47B36, 82B44
  • Retrieve articles in all journals with MSC (2010): 37A30, 47B36, 82B44
Bibliographic Information
  • Shiwen Zhang
  • Affiliation: Department of Mathematics, University of California Irvine, Irvine, California 92617
  • MR Author ID: 997713
  • Email: shiwez1@uci.edu
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 2603-2609
  • MSC (2010): Primary 37A30, 47B36, 82B44
  • DOI: https://doi.org/10.1090/proc/12929
  • MathSciNet review: 3477077