Inverse Scattering Problems Of The One-Dimensional Schr?dinger Equation With Magnetic Field

For the one-dimensional Schr?dinger equation with magnetic field:-f^??-2ipf^?-ipf=k²f,we study the inverse scattering problem.We prove the k space properties and Fourier properties of the scattering matrix S?k?.In addition,we prove that p can be uniquely determined by the reflection coefficient R?k?and obtain the trace formula of p.The prior upper bound of the solutions of the original equation is also obtained.This thesis is divided into four chapters.Chapter 1 introduces the background and significance of the inverse scattering problem of the Schr?dinger equation.Then we describe this kind of problems in detail and introduce its research status.In addition,the structure of this paper is introduced.Finally,we introduce the main symbols used in this paper.Chapter 2 introduces the form of the Schr?dinger equation and the condition of potential p.Furthermore,by transforming the solution function f₁?x,k?:m₁?x,k?=e^-??+ikx?f₁?x,k?,the k space properties and Fourier properties of m₁?x,k?and the scattering matrix S?k?are introduced.Finally,we introduce some related lemmas to prepare for the following.In Chapter 3,we prove that when the potential p makes the Schr?dinger operator be without bound states,it can be uniquely determined by the reflection coefficient R?k?.And we obtain the trace formula of the potential p based on R?k?.Finally,we obtain the priori upper bound of the solutions of the original equation.Chapter 4 summarizes the contents of this paper,and introduces that this paper only solves the inverse scattering problem of the original equation when the potential p makes the Schr?dinger operator be without bound states.Finally,it points out that the inverse scattering problem of the high dimensional?more than three dimensions?Schr?dinger equations are open problems.