The Quasi-periodic Schr(?)dinger Operators With C~2 Cos-type Potentials

We consider the one-dimensional discrete quasi-periodic Schr(?)dinger operator acting on l2(Z),H:l2(Z)→l2(Z)(Hα,λ,μ,xu)n:=un+1n+un-1+λv(x+nα)un,where v is a C2 COS-type potential function on R/Z,α is an arbitraly Diophantine frequency,λ>1 is a coupling constant.This paper can be divided into three topics as followIn the first part,we prove that the Lyapunov exponent(LE)of the corresponding Schodinger cocycle is 1/2-H(?)lder continuous as a function of the energy.Moreover,we prove the locally Lipschitz continuity of the LE for a full measure spectral set.Fur-thermore,for any given β between 1/2 and 1,we can find some energy on the spectrum and on which LE is between β-∈ and β+∈-H(?)lder continuous for any ∈>0.In the second part,We obtain the dry version of Cantor spectrum of the operatorIn the third part,we prove the absolute continuity of integrated density of states(IDS).We are going to state the structure of this paper.In the first chapter,we will first introduce the background and the previous results in the literature.Then,we present three main results of this paper in detail.In the second chapter,we will give some basic knowledge and the mathematics tools of our proof in this paper.In the first section,we introduce the relation between the Schr(?)dinger operators and Schodinger cocycles,the definition and the basic properties of the Lyapunov exponent,and the concept of the rotation number and the integral density of state.In the second section,we will introduce the classic large deviation theory and the avalanche principle.Next,we will introduce the techniques,developed by Wang and Zhang,to study the iteration of the Schr(?)dinger cocycle.By developing their technique,we give a key lemma,which play a key role in the following proof of the first result.In the third chapter,we give the proof of the local and global generality of LE.We divide the proof into several parts.At first,we introduce the the resonance and the classification of the spectrum and we show some topological properties for the spectrum.Secondly,according to the classification of the spectrum,we separately prove the local Lipschitz continuity of LE on a full-measure set(FR)and exactly 1/2-H(?)lder continuity on a zero-measure set(EP).Next,we deal with the regularity of LE on other energys.Finally,by the previous conclusions,we prove the global1/2-H(?)lder continuity of LE.In the forth chapter,combining the conclusion in[43]with our results given in the previous chapter,we turn the problem into directly calculating the rotation number on the gaps.The orbits are delicately estimated,which results in the proof of "dry"Cantor set(that is.each gap of the spectrum is open).Meanwhile,we give the upper bound and lower bound of each gap with the help of the previous techniques used in proving the regularity of LE.In the fifth chapter,combining the classification of the spectrum in the third chapter,the technique we used in the forth chapter and some classical results on the absolute continuity in real analysis,we finish the proof of the absolutely continuity of IDS.