Non-perturbative Generalizations Of Eliasson’s Reducibility Theory

This paper is concerned with the non-perturbative reducibility results of quasi-periodic linear systems based on Eliasson’s reducibility theory. The classic Eliasson’s theory showed a full measure reducibility result of quasi-periodic systems, but we remark that it is a perturbative result. Inspired by this result, this paper summarizes the reducibility results in continuous and discrete systems respectively and generalizes Eliasson’s theory non-perturbatively.This paper is mainly divided into four sections. Firstly, we introduce the back-ground and some basic knowledge of the reducibility, such as the integrated density of states, the rotation number and many other important concepts. In the discrete-time case, the paper use the analogous method of Puig and try to connect the Schrodinger operator and its dual operator by Aubry duality. Furthermore, we use the localization result of the dual operator to prove the existence of the Bloch wave and obtain the non-perturbative result finally. In the continuous-time situation, combined with Eliasson’s theory, we use the theory of almost reducibility to give a non-perturbative reducibility result for more general systems. Finally, we make a summary and prospect to give some more generalized results about the reducibility of quasi-periodic linear systems.