The Inverse Scattering Problem Of The Higher Order Schr?dinger Equation

The thesis is mainly concerned of the inverse scattering problems of the higher order Schr ?dinger equation.First we prove a uniqueness result of this problem by the method of constructing the complex geometrical optics solutions.Moreover,we also prove that the partial scattering data can also determines the potentials locally,i.e.,the local uniqueness of the inverse backscattering problem,which we improve the result of the inverse backscattering problem of the second order Schr ?dinger equation.This thesis is divided into five chapters.In Chapter 1,we introduce the physical background of Schr ?dinger equation as well as the inverse scattering problems,and introduces the research status of this problem in the case of the second order Schr ?dinger equation,then presents the main results of the thesis.In Chapter 2,we discuss the inverse scattering problem of the higher order Schr ?dinger equation with the first order perturbation potential.Our method is based on the relation between the asymptotic expansion of the resolvent and the scattering matrix.Under the condition that the potential has exponential decay,we make a expansion of the resolvent and derivatives of the resolvent of the higher order Schr ?dinger operators.Then we construct the Poisson operator and establish the relationship between potential and scattering matrix by using the progressive expansion made before.Finally,we construct the complex geometric optics solution of the higher order Schr ?dinger equation,and bring in the relation between potential and scattering matrix,so that we can prove that the scattering matrix can determines the potential uniquely.In Chapter 3,we consider the inverse backscattering problem of the second order Schr ?dinger equation with noncompact support potential,and prove that the potential with some decaying can be determined locally and uniquely by the backscattering amplitude.For the higher order Schr ?dinger equation,we also prove the corresponding results under appropriate conditions.In Chapter 4,we consider the Strichartz estimation of the higher order Schr ?dinger equation,and generalize some known results for the second order cases.In the final Chapter 5,a summary is presented and further questions are also discussed.