Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-13T10:47:48.227Z Has data issue: false hasContentIssue false
  • English
  • Français

Pure point spectrum for the Maryland model: a constructive proof

Published online by Cambridge University Press:  09 August 2019

SVETLANA JITOMIRSKAYA  and
FAN YANG Open the ORCID record for FAN YANG [Opens in a new window]
SVETLANA JITOMIRSKAYA
Affiliation:
University of California, Department of Mathematics, Irvine, California, USA email szhitomi@math.uci.edu
FAN YANG
Affiliation:
Georgia Institute of Technology, School of Mathematics, Atlanta, Georgia, USA email ffyangmath@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a constructive method to prove and study pure point spectrum for the Maryland model with Diophantine frequencies.

Keywords

quasiperiodic operatorsAnderson localizationDiophantine
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 1 , January 2021 , pp. 283 - 294
Check for updates
Copyright
© Cambridge University Press, 2019

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.)

References

Avila, A. and Jitomirskaya, S.. The ten Martini problem. Ann. of Math. (2) 170 (2009), 303–342.Google Scholar
Berezansky, Y. M.. Expansions in Eigenfunctions of Selfadjoint Operators. American Mathematical Society, Providence, RI, 1968.Google Scholar
Berry, M.. Incommensurability in an exactly-soluble quantal and classical model for a kicked rotator. Phys. D 10 (1984), 369–378.Google Scholar
Fedotov, A. and Sandomirskiy, F.. An exact renormalization formula for the Maryland model. Comm. Math. Phys. 334(2) (2015), 1083–1099.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.Google Scholar
Fishman, S.. Anderson localization and quantum chaos maps. Scholarpedia 5(8) (2010), 9816.Google Scholar
Furman, A.. On the multiplicative ergodic theorem for uniquely ergodic systems. Ann. Inst. Henri Poincaré Probab. Stat. 33(6) (1997), 797–815.Google 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(4) (2014), 041405.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.Google Scholar
Han, R.. Shnol’s theorem and the spectrum of long range operators. Proc. Amer. Math. Soc. 147(7) (2019), 2887–2897.Google Scholar
Han, R., Jitomirskaya, S. and Yang, F.. Universal hierarchical structure of eigenfunctions in the Maryland model, in preparation.Google Scholar
Jitomirskaya, S.. Metal-insulator transition for the almost Mathieu operator. Ann. of Math. (2) 150(3) (1999), 1159–1175.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(1) (2005), 103–124.Google Scholar
Jitomirskaya, S. and Liu, W.. Arithmetic spectral transitions for the Maryland model. Comm. Pure Appl. Math. 70(6) (2017), 1025–1051.Google Scholar
Jitomirskaya, S. and Liu, W.. Universal hierarchical structure of quasiperiodic eigenfunctions. Ann. of Math. (2) 187(3) (2018), 721–776.Google Scholar
Jitomirskaya, S. and Liu, W.. Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transitions in phase. Preprint, 2018, arXiv:1802.00781.Google Scholar
Jitomirskaya, S., Liu, W. and Tcheremchantzev, S.. Wavepacket spreading and fractal spectral dimension of quasiperiodic operators with singular continuous spectrum, in preparation.Google Scholar
Jitomirskaya, S. and Mavi, R.. Dynamical bounds for quasiperiodic Schrödinger operators with rough potentials. Int. Math. Res. Not. IMRN 1 (2017), 96–120.Google Scholar
Liu, W. and Yuan, X.. Anderson Localization for the almost Mathieu operator in exponential regime. J. Spectr. Theory 5(1) (2015), 89–112.Google Scholar
Prange, R., Grempel, D. and Fishman, S.. Wave functions at a mobility edge: an example of a singular continuous spectrum. Phys. Rev. B 28(12) (1983), 7370.Google Scholar
Simon, B.. Almost periodic Schrödinger operators. IV. The Maryland model. Ann. Phys. 159(1) (1985), 157–183.Google Scholar