Nonlinear Scattering Theory For Schr(?)dinger Equation With A Potential In H~s

There are various phenomena in nature (like the blue of the sky), Scattering theory is from quantum mechanics, it is the most effective means, sometimes the only means, to study microscopic nature, so it involves very widely. One often wants to use the scattering of waves or particles whose dynamics one knows to determine the structure and position of small or inaccessible objects. For example, in X-ray crystallography (which led to the discover of DNA), tomography, and the detection of underwater objects by sonar, the underlying dynamics is well understood. Scattering theory also plays very important roles in dynamics . In elementary particle physics, essentially all the experimental data are scattering data,the main test of any proposed particle dynamics is whether one can construct for the dynamics a scattering theory that predicts the observed experimental data.In this paper we first introduce the developing history of Schr?dinger equation and physical background of scattering phenomenon,then give a simple introduction to different means for Schr?dinger equation with potential, and give some conclusions for developing history and approach research. On the basis of other's excellent achievement, we will discuss the scattering theory for nonlinear Schr?dinger equation with potential in H s. In section two, we introduce the definition of wave operator and scattering operator, and their proof of existence. In section three, we discuss the scattering theory for nonlinear Schr?dinger equation without potential by treating nonlinearity in Besove space. In section three, we discuss the scattering theory for nonlinear Schr?dinger equation with potential by treating nonlinearity in Besove space and the spatial decay of solution is obtained. Conclusions and developing foreground are provided in section five.