Scattering For One Class Of Schr(?)dinger Equation In The Low Regularity

Estimates of Schr(o|¨)dinger equation is one of the open questions in mathematicalfield,especially nonlinear Schr(o|¨)dinger equation with potential. Research of initial valueproblem on this kind of equation has attracted much attention of many scientists all overthe world,because study on this kind of equation has both very important theoreticalsignificance and abundant application background.There have been a lot of conclusionson estimates of Schr(o|¨)dinger equation with potential, and these lay a foundation forfurther research.The primary purpose of this paper is to investigate the asymptotic completenessand scattering of nonlinear Schr(o|¨)dinger equation with di?erent kinds of potentials byTao's I-energy method and gains general conclusions. In this paper we add di?erentkinds of potentials,and we talk about the low reglarity in the case of s < 1.This makesthe problem complicated and we get similar conclusions.This paper compose of three chapters.In the first chapter,we introduce the estimatesof Schr(o|¨)dinger equation by reviewing the basic knowledge on Schr(o|¨)dinger equation andoverviews the main conclusions of this paper.In the second chapter, we introduce the mainpropositions and knowledge cited in this paper.The third chapter changes the potentialof Schr(o|¨)dinger equation into iA ?? + V (x, t)?,restricts A and V (x, t) with severalconditions,obtains Lx4,t estimates on the solution of initial value problem with potentialiA ??+V (x, t)? and we get the conclusion that energy nearly conserves.Then the paperobtains the conclusions of asymptotic completeness and the existence of wave operator inthis form.And when A = 0,the potential of Schr(o|¨)dinger equation is changed into V (x, t)?and we gets the similar conclusions with the same method. And the argumentation inthis case is much easier than before.