The Lyapunov Exponents And Topological Structures Of Spectrum For Finitely Smooth Quasi-periodic Schr(?)dinger Operators

We consider the one-dimensional discrete quasi-periodic Schr(?)dinger operator act-ing on l2(Z),where v is a C2 cos-type potential function on R/Z,ca is an arbitrary irrational fre-quency,A is a coupling constant.We will re-obtain the continuity of the Lyapunov exponents without a large deviation theorem for the Schr(?)dinger operators with a sat-isfying Diophantine condition when ? is sufficiently large.On the other hand,for each fixed weak Liouville frequency ?,we prove that the spectrum of the corresponding Schr(?)dinger operator is a Cantor sot for sufriciontly largo ?.In chapter one,we will introduce the background of Schr(?)dinger operators and the significance of research for Lyapunov exponents and Cantor spectrum.Then we will present some basic knowledge which is related to our main results in this thesis such as the definitions of C2 cos-type potential functions,Lyapunov exponents,uniformly hyperbolic system and Cantor spectrum.We will just give simple presentation for some content without any proof because they are well-known.In chapter two,we will give a new proof of continuity of Lyapunov exponents with Diophantine frequencies.We will first present the research progress of continuity of Lyapunov exponents and show our main result of this chapter.Then we will give a key technical definition and estimate on the convergent speed of finite Lyapunov exponents Finally,we prove the main result of this chapter.In chapter three,we will prove that the spectrum of the responding Schr(?)dinger operator,with weak Liouville frequencies,is a Cantor set for sufficiently large A.We will first present the research progress of Cantor spectrum and show our main result in this chapter.Then we will give some important lemmas and the strategy of the proof of the main result in this chapter.Then we will give some review of Liang-Kung[33],particularly we will make an detailed analysis of the method to the main result of[33].Finally,we will give the proof of the main result in this chapter by proving that uniformly hyperbolic systems are dense.