Polynomial decay of the gap length for Ck quasi-periodic Schrodinger operators and spectral application

For the quasi-periodic Schrodinger 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 SL(2, R) cocycles, and also based on the Moser-Poschel argument for the related Schrodinger cocycles. As an application, we are able to show the homogeneity of the spectrum. (C) 2021 Elsevier Inc. All rights reserved.