Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent


Authors: Valmir Bucaj, David Damanik, Jake Fillman, Vitaly Gerbuz, Tom VandenBoom, Fengpeng Wang and Zhenghe Zhang
Journal: Trans. Amer. Math. Soc. 372 (2019), 3619-3667
MSC (2010): Primary 35J10; Secondary 81Q10
DOI: https://doi.org/10.1090/tran/7832
Published electronically: April 29, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by Fürstenberg's theorem. That is, a Schrödinger operator in $ \ell ^2(\mathbb{Z})$ whose potential is given by independent, identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions, and its unitary group exhibits exponential off-diagonal decay, uniformly in time. We also explain how to obtain analogous statements for extended CMV matrices whose Verblunsky coefficients are i.i.d., as well as for half-line analogues of these models.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35J10, 81Q10

Retrieve articles in all journals with MSC (2010): 35J10, 81Q10


Additional Information

Valmir Bucaj
Affiliation: Department of Mathematics, United States Military Academy, West Point, New York 10996
Email: valmir.bucaj@westpoint.edu

David Damanik
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005
Email: damanik@rice.edu

Jake Fillman
Affiliation: Department of Mathematics, Virginia Tech, 225 Stanger Street—0123, Blacksburg, Virginia 24061
Email: fillman@vt.edu

Vitaly Gerbuz
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005
Email: vitaly.gerbuz@rice.edu

Tom VandenBoom
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
Email: thomas.vandenboom@yale.edu

Fengpeng Wang
Affiliation: School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, Guangdong 519082, People’s Republic of China
Email: wfpouc@163.com

Zhenghe Zhang
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: zhenghe.zhang@ucr.edu

DOI: https://doi.org/10.1090/tran/7832
Received by editor(s): November 30, 2018
Received by editor(s) in revised form: February 19, 2019
Published electronically: April 29, 2019
Additional Notes: The first, second, fourth, and fifth authors were supported in part by NSF grant DMS-1361625.
The main idea of the new proof of the LDT in Section 3 was communicated to the second and seventh authors by Artur Avila while they were visiting IMPA, Rio de Janeiro. They would like to thank Artur Avila for sharing his idea, and IMPA for the hospitality.
The third author was supported in part by an AMS-Simons travel grant, 2016–2018
The sixth author was supported by CSC (No. 201606330003) and NSFC (No. 11571327).
The seventh author was supported in part by an AMS-Simons travel grant, 2014–2016
Article copyright: © Copyright 2019 by the authors