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arXiv · 2006.15347

Polynomial decay of the gap length for C^k quasi-periodic Schrodinger operators and spectral application

Ao Cai·Xueyin Wang
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Abstract

For the quasi-periodic Schrödinger operators in the local perturbative regime where the frequency is Diophantine and the potential is $C^k$ sufficiently small depending on the Diophantine constants, we prove that the length of the corresponding spectral gap has a polynomial decay upper bound with respect to its label. This is based on a refined quantitative reducibility theorem for $C^k$ quasi-periodic ${\rm SL}(2,\mathbb{R})$ cocycles, and also based on the Moser-Pöschel argument for the related Schrödinger cocycles. As an application, we are able to show the homogeneity of the spectrum.

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Ao Cai, Xueyin Wang. 2020-06-27. Polynomial decay of the gap length for C^k quasi-periodic Schrodinger operators and spectral application. https://arxiv.org/abs/2006.15347

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