Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-06T06:21:30.535Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral Aspects of the Skew-Shift Operator: A Numerical Perspective

Published online by Cambridge University Press:  03 June 2015

Eric Bourgain-Chang
Eric Bourgain-Chang*
Affiliation:
Mechanical Engineering Department, University of California, Berkeley, CA 94720, USA
*
*Corresponding author.Email:[email protected]
Get access

Abstract

In this paper we perform a numerical study of the spectra, eigenstates, and Lyapunov exponents of the skew-shift counterpart to Harper’s equation. This study is motivated by various conjectures on the spectral theory of these ‘pseudo-random’ models, which are reviewed in detail in the initial sections of the paper. The numerics carried out at different scales are within agreement with the conjectures and show a striking difference compared with the spectral features of the Almost Mathieu model. In particular our numerics establish a small upper bound on the gaps in the spectrum (conjectured to be absent).

Keywords

81Q1039A7047B39Schrödinger operatorskew-shiftspectrumlocalization
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 3 , March 2014 , pp. 712 - 732
Copyright
Copyright © Global Science Press Limited 2014

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

[1]Avila, A., Jitomirskaya, S., The ten Martini problem, Annals of Math 170 (2009), 303–342).Google Scholar
[2]Brenner, N., Fishman, S., Pseudo-randomness and localization, Nonlinearity 4 (1992), 211–235.Google Scholar
[3]Figotin, A., Pastur, L., Spectra of random and almost periodic operators, Springer-Verlag, 1992.Google Scholar
[4]Griniasty, M., Fishman, S., Localization by pseudorandom potentials in one dimension, Phys. Rev, Lett, 60, 1334–1337 (1988).CrossRefGoogle ScholarPubMed
[5]Gordon, A., Jitomirskaya, S., Last, Y., Simon, B., Duality and singular continuous spectrum in the almost Mathieu equation, Acta Math. 178 (1997), 202, 169–183.Google Scholar
[6]Jitomirskaya, S.Y., Metal-insulator transition for the almost Mathieu operator, Ann. Math. (2) 150(3), 1159–1175 (1999).CrossRefGoogle Scholar
[7]Jitomirskaya, S., Krasovsky, I.V., Continuity of the measure of the spectrum for discrete quasi-periodic operators, Math. Res. Letters 9 (2002), 413–421.Google Scholar
[8]Krüger, H., The spectrum of skew-shift Schrödinger operator contains intervals, Journal of Funct. Anal. 262 (2012), 203, 773–810.Google Scholar
[9]Krüger, H., An explicit example of a skew-shift Schrödinger operator with positive Lyapunov exponent at small coupling, arXiv:1206:1362.Google Scholar
[10]Krüger, H., A family of Schrödinger Operators whose spectrum is an interval, Comm. Math. Phys. 290:3, 935–939 (2009).Google Scholar
[11]Krüger, H., Probabilistic averages of Jacobi operators, Comm. Math. Phys. 295:3 (2010).CrossRefGoogle Scholar
[12]Last, Y., Almost everything about the almost Mathieu operator. I, In: XIth International Congress of Mathematical Physics, Paris, 1994, pp. 366–372.Google Scholar
[13]Last, Y., Simon, B., Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999), 329–367.Google Scholar
[14]Peierls, R., Zur theorie des Diamagnetismus von Leitungselektronen, Phys, Z. 80 (1933), 763–791.Google Scholar
[15]Schlag, W., Private Communication.Google Scholar