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

2020/06/27 by Cai, Ao, Wang, Xueyin
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2006.15347

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,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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