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

TL;DR

This paper proves that spectral gaps in certain quasi-periodic Schrödinger operators decay polynomially with their label, using advanced reducibility techniques, and demonstrates the spectrum's homogeneity as an application.

Contribution

It introduces a refined reducibility theorem for $C^k$ quasi-periodic cocycles and applies it to establish polynomial decay of spectral gaps and spectrum homogeneity.

Findings

Spectral gap lengths decay polynomially with their label.

The spectrum of the operator is shown to be homogeneous.

The results depend on the potential being sufficiently small and the frequency being Diophantine.

Abstract

For the quasi-periodic Schr\"{o}dinger 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-P\"{o}schel argument for the related Schr\"{o}dinger cocycles. As an application, we are able to show the homogeneity of the spectrum.