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arXiv · 1712.09041

Reducibility of finitely differentiable quasi-periodic cocycles and its spectral applications

Ao Cai·Lingrui Ge
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Abstract

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 of multi-frequency $C^k$ long-range operators on $\ell^2(\Z^d)$. These results are based on reducibility properties of finitely differentiable quasi-periodic $SL(2,\R)$ cocycles. More precisely, we prove that if the base frequency is Diophantine, then a $C^k$ $SL(2,\R)$-valued cocycle is reducible if it is close to a constant cocycle, sufficiently smooth and the rotation number of it is Diophantine or rational with respect to the frequency.

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Ao Cai, Lingrui Ge. 2017-12-25. Reducibility of finitely differentiable quasi-periodic cocycles and its spectral applications. https://arxiv.org/abs/1712.09041

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