# $L^2$-reducibility and localization for quasiperiodic operators

@article{Jitomirskaya2015L2reducibilityAL,
title={\$L^2\$-reducibility and localization for quasiperiodic operators},
author={Svetlana Ya. Jitomirskaya and Ilya Kachkovskiy},
journal={arXiv: Spectral Theory},
year={2015}
}
• Published 26 May 2015
• Mathematics
• arXiv: Spectral Theory
We give a simple argument that if a quasiperiodic multi-frequency Schr\"odinger cocycle is reducible to a constant rotation for almost all energies with respect to the density of states measure, then the spectrum of the dual operator is purely point for Lebesgue almost all values of the ergodic parameter $\theta$. The result holds in the $L^2$ setting provided, in addition, that the conjugation preserves the fibered rotation number. Corollaries include localization for (long-range) 1D analytic…
24 Citations
Reducibility of Finitely Differentiable Quasi-Periodic Cocycles and Its Spectral Applications
• Mathematics
• 2017
In this paper, we prove the generic version of Cantor spectrum for quasi-periodic Schr\"{o}dinger operators with finitely smooth and small potentials, and we also show pure point spectrum for a class
Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,R)$ cocycles
• Mathematics
• 2020
We prove that if the frequency of the quasi-periodic SL(2,R) cocycle is Diophantine, then the following properties are dense in the subcritical regime: for any 1 2 < κ < 1, the Lyapunov exponent is
HÖLDER REGULARITY OF THE INTEGRATED DENSITY OF STATES FOR QUASI-PERIODIC LONG-RANGE OPERATORS ON 2(Zd)
• Mathematics
• 2020
We prove the Hölder continuity of the integrated density of states for a class of quasiperiodic long-range operators on (Z) with large trigonometric polynomial potentials and Diophantine
On the relation between strong ballistic transport and exponential dynamical localization
We establish strong ballistic transport for a family of discrete quasiperiodic Schr\"odinger operators as a consequence of exponential dynamical localization for the dual family. The latter has been,
H\"older regularity of the integrated density of states for quasi-periodic long-range operators on $\ell^2(\Z^d)$
• Mathematics
• 2020
We prove the Holder continuity of the integrated density of states for a class of quasi-periodic long-range operators on $\ell^2(\Z^d)$ with large trigonometric polynomial potentials and Diophantine
Arithmetic version of anderson localization for quasiperiodic Schr\"odinger operators with even cosine type potentials
• Mathematics
• 2021
We propose a new method to prove Anderson localization for quasiperiodic Schrodinger operators and apply it to the quasiperiodic model considered by Sinai and Frohlich-Spencer-Wittwer. More
Universal hierarchical structure of quasiperiodic eigenfunctions
• Mathematics
• 2016
We determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime. This uncovers a
Arithmetic version of Anderson localization via reducibility
• Mathematics, Computer Science
• 2020
A novel approach based on an arithmetic version of Aubry duality and quantitative reducibility is proposed to prove the same result for the class of quasi-periodic long-range operators in {\it all dimensions}, which includes \cite{J, bj02} as special cases.
QUANTITATIVE ALMOST REDUCIBILITY AND ITS APPLICATIONS
• J. You
• Mathematics
Proceedings of the International Congress of Mathematicians (ICM 2018)
• 2019
We survey the recent advances of almost reducibility and its applications in the spectral theory of one dimensional quasi-periodic Schrödinger operators. 1 Quasi-periodic operators, cocycles and

## References

SHOWING 1-10 OF 40 REFERENCES
A nonperturbative Eliasson's reducibility theorem
This paper is concerned with discrete, one-dimensional Schrodinger operators with real analytic potentials and one Diophantine frequency. Using localization and duality we show that almost every
Reducibility or nonuniform hyperbolicity for quasiperiodic Schrodinger cocycles
• Mathematics
• 2003
We show that for almost every frequency ?? ?? R\Q, for every C?O potential v : R/Z ?? R, and for almost every energy E the corresponding quasiperiodic Schr?Nodinger cocycle is either reducible or
Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems
• Mathematics
• 2012
In this paper, we prove that a quasi-periodic linear differential equation in sl(2,ℝ) with two frequencies (α,1) is almost reducible provided that the coefficients are analytic and close to a
Almost localization and almost reducibility
• Mathematics
• 2008
We develop a quantitative version of Aubry duality and use it to obtain several sharp estimates for the dynamics of Schr\"odinger cocycles associated to a non-perturbatively small analytic potential
A KAM scheme for SL(2,R) cocycles with Liouvillean frequencies
• Mathematics
• 2010
We develop a new KAM scheme that applies to SL(2,R) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg-Sinai's Theorem
Almost reducibility and absolute continuity I
We consider one-frequency analytic SL(2,R) cocycles. Our main result establishes the Almost Reducibility Conjecture in the case of exponentially Liouville frequencies. Together with our earlier work,
Absence of localisation in the almost Mathieu equation
The author considers the discrete Schrodinger operator on Z with the potential lambda cos 2 pi ( alpha n+ theta ). This one-dimensional model occurs in the study of an electron in a two-dimensional
A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants
We study ergodic Jacobi matrices onl2(Z), and prove a general theorem relating their a.c. spectrum to the spectra of periodic Jacobi matrices, that are obtained by cutting finite pieces from the
Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation
We show that the 1-dimensional Schrödinger equation with a quasiperiodic potential which is analytic on its hull admits a Floquet representation for almost every energyE in the upper part of the