Anderson Localization In Disordered Dynamical Systems

In this thesis, we try to explain and investigate Anderson localization, an intriguing physical phenomenon, from the perspective of mathematics. The disordered systems we consider are two quasi-crystal models, i.e.,· one-dimensional nonlinear Maryland model: iqn=∈(qn+1+qn-1)+tanπ(x+nα)qn+∈|qn|2qn, n∈Z,(0.3)where x∈R/Z, and α∈Rd is some fixed Diophantine number;· one-dimensional nonlinear quasi-periodic Schrodinger equation: iqn=∈(qn+1+qn-1)+V(x+nα)qn+|qn|2qn, n∈Z,(0.4)where V is a nonconstant real-analytic function on R/Z, and a is some fixed Diophantine number.In the first chapter, we take the ergodic Schrodinger operator as the main object of study, to explain localization in linear disorder systems. Some concepts in the spectral theory of operators, e.g., exponential localization, dynamical localization, will be given in this chapter. For three significant models, i.e., linear Anderson model, linear Maryland model and one-dimensional linear quasi-periodic Schrodinger operator, we shall state the corresponding conclusions respectively. In the second chapter, we consider the one-dimensional nonlinear Maryland mod-el. We shall prove that, for "most" compactly-supported small-amplitude initial data (qn(0))n∈Z，if∈is sufficiently small, then for "most" x∈R/Z, the solution (qn(t))n∈Z of Equation (0.3) satisfies:(?)s>0, the diffusion norm is uniformly bounded with respect to t.In the third chapter, we consider the one-dimensional nonlinear quasi-periodic Schrodinger equation. For "most" compactly-supported initial data (qn(0))n∈Z, if∈is sufficiently small, then for a.e. x∈R/Z, the solution (qn(t))n∈Z of Equation (0.4) satisfies:(?)s>0,...