CaltechAUTHORS
  A Caltech Library Service

Analytic Quasi-Perodic Cocycles with Singularities and the Lyapunov Exponent of Extended Harper’s Model

Jitomirskaya, S. and Marx, C. A. (2012) Analytic Quasi-Perodic Cocycles with Singularities and the Lyapunov Exponent of Extended Harper’s Model. Communications in Mathematical Physics, 316 (1). pp. 237-267. ISSN 0010-3616. https://resolver.caltech.edu/CaltechAUTHORS:20130214-075409144

Full text is not posted in this repository. Consult Related URLs below.

Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20130214-075409144

Abstract

We show how to extend (and with what limitations) Avila’s global theory of analytic SL(2,C) cocycles to families of cocycles with singularities. This allows us to develop a strategy to determine the Lyapunov exponent for the extended Harper’s model, for all values of parameters and all irrational frequencies. In particular, this includes the self-dual regime for which even heuristic results did not previously exist in physics literature. The extension of Avila’s global theory is also shown to imply continuous behavior of the LE on the space of analytic M_2 (C)-cocycles. This includes rational approximation of the frequency, which so far has not been available.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/s00220-012-1465-4DOIArticle
http://link.springer.com/article/10.1007/s00220-012-1465-4PublisherArticle
http://dx.doi.org/10.1007/s00220-012-1637-2 DOIErratum
Additional Information:© 2012 Springer-Verlag. Received: 14 September 2011; Accepted: 27 October 2011; Published online: 31 March 2012. The work was supported by the NSF Grants DMS-0601081 and DMS-1101578, and the BSF grant 2006483. Communicated by B. Simon. We are grateful to Artur Avila for his remarks on an earlier version of this paper where, among other things, he essentially provided a simple proof of continuity of the Lyapunov exponent of singular cocycles for all frequencies once such continuity for the Diophantine case is established, using the ideas of [2]. It turned out the same idea could be used to provide a simple proof of joint continuity (presented here), significantly simplifying our original approach. Additionally, our proof of continuity for the case of an identically vanishing determinant follows his suggestions as well.We also thank Anton Gorodetski for useful discussions during the preparation of this manuscript.
Funders:
Funding AgencyGrant Number
NSFDMS-0601081
NSFDMS-1101578
Binational Science Foundation (BSF)2006483
Issue or Number:1
Record Number:CaltechAUTHORS:20130214-075409144
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20130214-075409144
Official Citation:Jitomirskaya, S. and C. A. Marx (2012). "Analytic Quasi-Perodic Cocycles with Singularities and the Lyapunov Exponent of Extended Harper’s Model." Communications in Mathematical Physics 316(1): 237-267.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:36907
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:26 Feb 2013 22:39
Last Modified:03 Oct 2019 04:42

Repository Staff Only: item control page