The absolutely continuous spectrum of finitely differentiable quasi-periodic Schr\"{o}dinger operators

TL;DR

This paper proves that finitely differentiable quasi-periodic Schrödinger operators with small potentials and Diophantine frequencies have purely absolutely continuous spectra, extending spectral theory to lower regularity potentials.

Contribution

It introduces a refined almost reducibility theorem applicable to finitely differentiable potentials with low initial regularity, ensuring absolutely continuous spectrum.

Findings

Absolutely continuous spectrum for small potentials

Refined almost reducibility theorem for finitely differentiable potentials

Spectral results hold under low regularity conditions

Abstract

We prove that the quasi-periodic Schr\"{o}dinger operator with a finitely differentiable potential has purely absolutely continuous spectrum for all phases if the frequency is Diophantine and the potential is sufficiently small in the corresponding $C^k$ topology. This is based on a refined quantitative $C^{k,k_0}$ almost reducibility theorem which only requires a quite low initial regularity ``$k>14\tau+2$'' and much of the regularity ``$k_0\leq k-2\tau-2$'' is conserved in the end, where $\tau$ is the Diophantine constant of the frequency.