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

TL;DR

This paper establishes a link between $L^2$-reducibility of quasiperiodic Schrödinger cocycles and spectral localization, showing that reducibility implies pure point spectrum for the dual operator under certain conditions.

Contribution

It provides a simple argument connecting reducibility to localization in the $L^2$ setting, extending results to multidimensional cases and long-range potentials.

Findings

Reducibility to a constant rotation implies pure point spectrum for the dual operator.

Localization results for 1D analytic potentials with dual absolutely continuous spectrum.

New multidimensional localization results.

Abstract

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 potentials with dual ac spectrum and Diophantine frequency as well as a new result on multidimensional localization.