Reducibility And The Spectrum Of Quasi-periodic Schr(?)dinger Operators

This thesis is devoted to study the reducibility of quasi-periodic linear system in sl(2,R).Meanwhile,quasi-periodic Schr(?)dinger operator and its characteristic equation provide an important example which can be transformed into such a system,and has been intensively studied in last decades.We review some remarkable results about reducibility,almost reducibility,and as an application,we explore the connections between reducibility,almost reducibility and the spectrum of quasi-periodic Schrodinger operator.In the meantime,this thesis gives another proof of the rotation reducibility[32]of positive measure of rotation number under two frequencies w=(?,1).The proof is based on CD-bridge technique introduced by Avila,Fayad,and Krikorian[10],see Theorem3.5 for more details.In the first chapter,we introduce the subjects of this thesis:quasi-periodic linear system and Schrodinger operator.Then we explain some fundamental definitions and concepts,such as reducibility,rotation number,Diophantine condition,etc.Furthermore,we give some preliminaries which are useful in the following chapters.In the second chapter,we deal with the quasi-periodic linear system originated from quasi-periodic Schr(?)dinger characteristic equation with eigenvalues ? in the resolvent set of the corresponding quasi-periodic Schrodinger operator.It turns out that,in the resolvent set,the quasi-periodic linear system is always reducible and it can be reduced to uniformly hyperbolic case.In the third chapter,we use a more systematic method to study local reducibility.The main technique is KAM theory.KAM theory provide an approach to study the reducibility in the spectrum.And reducibility in the spectrum implies absolutely continuous spectrum.While impose Diophantine condition to the system,local reducibility can be divided into two cases,perturbation reducibility depending on the Diophantine constants,and non-perturbation reducibility not depending on the Diophantine constants.One can not expect reducibility anymore under Liouvillean condition,however,almost reducibility and rotation reducibility is reasonable,which still have many applications in the study of spectrum.In the fourth chapter,we sketch some latest progresses of applications of quantitative almost reducibility in the spectrum.In the spectral set,we concern about Cantor spectrum,estimate of spectrum gap and homogeneous spectrum.While in the spectral measure,we are interested in Anderson localization and absolutely continuous spectrum.The last chapter is appendix.We give a proof of Floquet theorem which is a good complement to the thesis.