$L^2$-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… 
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TLDR
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…
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