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 Mosc. Math. J., 2005, Volume 5, Number 3, Pages 577–612 (Mi mmj211)

On Schrödinger operators with dynamically defined potentials

M. Sh. Goldsteina, W. Schlagb

a Department of Mathematics, University of Toronto
b University of Chicago

$$(H_\psi)_n=-\psi_{n-1}-\psi_{n+1}+\lambda V(T^n x)\psi_n$$
on $\ell^2(\mathbb Z)$, where $T\colon X\to X$ is an ergodic transformation on $(X,\nu)$ and $V$ is a real-valued function. $\lambda$ is a real parameter called coupling constant. Typically, $X=\mathbb T^d=(\mathbb R/\mathbb Z)^d$ with Lebesgue measure, and $V$ will be a trigonometric polynomial or analytic. We shall focus on our earlier papers, as well as other work which was obtained jointly with Jean Bourgain. Our goal is to explain some of the methods and results from these references. Some of the material in this paper has not appeared elsewhere in print.

Key words and phrases: Eigenfunction, localization, Lyapunov exponent.

DOI: https://doi.org/10.17323/1609-4514-2005-5-3-577-612

Full text: http://www.ams.org/.../abst5-3-2005.html
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MSC: 47B80
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Citation: M. Sh. Goldstein, W. Schlag, “On Schrödinger operators with dynamically defined potentials”, Mosc. Math. J., 5:3 (2005), 577–612

Citation in format AMSBIB
\Bibitem{GolSch05} \by M.~Sh.~Goldstein, W.~Schlag \paper On Schr\"odinger operators with dynamically defined potentials \jour Mosc. Math.~J. \yr 2005 \vol 5 \issue 3 \pages 577--612 \mathnet{http://mi.mathnet.ru/mmj211} \crossref{https://doi.org/10.17323/1609-4514-2005-5-3-577-612} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2241813} \zmath{https://zbmath.org/?q=an:1143.47301} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208595500007} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Goldstein M., Schlag W., “On the formation of gaps in the spectrum of Schrödinger operators with quasi-periodic potentials”, Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday - ERGODIC SCHRODINGER OPERATORS, SINGULAR SPECTRUM, ORTHOGONAL POLYNOMIALS, AND INVERSE SPECTRAL THEORY, Proceedings of Symposia in Pure Mathematics, 76, no. 2, 2007, 591–611
2. Goldstein M., Schlag W., “On resonances and the formation of gaps in the spectrum of quasi-periodic Schrodinger equations”, Ann of Math (2), 173:1 (2011), 337–475
3. Krueger H., “Multiscale Analysis for Ergodic Schrodinger Operators and Positivity of Lyapunov Exponents”, J Anal Math, 115 (2011), 343–387
4. Duarte P., Klein S., “Large Deviation Type Estimates For Iterates of Linear Cocycles”, Stoch. Dyn., 16:3, SI (2016), 1660010