# $L^2$-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} }

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

- Mathematics
- 2020

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…

Full measure reducibility and localization for quasiperiodic Jacobi operators: A topological criterion

- Mathematics, Physics
- 2017

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

- MathematicsProceedings 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…

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