The Applications Of Almost Reducibility In Quasiperiodic Schr(?)dinger Operators

Quasiperiodic Schrodinger operator is a mathematical model of quantum Hall effect and many other quantum physical problems.It is an important research topic in mathematical physics.The quasiperiodic Schr(?)dinger operator is closely related to the quasiperiodic Schrodinger cocycle which is a special cocycle.Almost reducibility is a very important method derived from applying the famous KAM theory to quasiperiodic cocycle.In this paper,we study the spectral theory of quasiperiodic Schr(?)dinger operators based on the almost reducibility,which belongs to the cross subject of dynamical systems and spectral theory.Our main content is divided into the following two parts:One is to transform the study of spectral problems related to various Schr(?)dinger operators into the study of asymptotic behavior of the n-th iterates of Schr(?)dinger cocycle,that is to build a bridge between spectral theory and dynamical systems.The other is to use the quantitative almost reducibility to describe the asymptotic behavior of the n-th iterates of Schr(?)dinger cocycle,that is,to establish the relationship between the quantitative almost reducibility estimation and the asymptotic behavior of the iterates of cocycle.This paper consists of nine chapters.The first two chapters are the introduction and preliminary.The last chapter is the summary and prospect.Chapter 3 to Chapter 8 are the main parts of this article.The details are as follows:In the third chapter,we use the full measure reducibility for smooth SL(2,R)cocycle to study and partially solve the open problem proposed by Jitomirskaya;In Chapter 4,we use the full measure reducibility for smooth SL(2,R)cocycle to study and partially solve Last's intersection spectrum conjecture;In Chapter 5,we study and prove the regularity of the spectral measures based on the quantitative almost reducibility for analytic SL(2,R)cocycle;In Chapter 6,we study and obtain the estimation of spectral gaps and homogeneous spectrum based on the reducibility for analytic SL(2,R)cocycle;In Chapter 7,we give a new proof of the famous Aubry-André-Jitomirskaya phase transition conjecture based on the full measure reducibility for analytic SL(2,R)cocycle;In Chapter 8,we study and prove the regularity of the Lyapunov exponent based on the quantitative almost reducibility for high dimensional analytic cocycle.