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arXiv · 2103.15525

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

Ao Cai
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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.

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Ao Cai. 2021-03-29. The absolutely continuous spectrum of finitely differentiable quasi-periodic Schr\"{o}dinger operators. https://arxiv.org/abs/2103.15525

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