【作者】 黄华；

【导师】 郑权；

【作者基本信息】 华中科技大学 ， 应用数学， 2019， 博士

【摘要】 本文主要研究高阶Schr?dinger方程的逆散射问题.首先我们利用构造复几何光学解的方法证明了高阶Schr?dinger方程逆散射问题的唯一性结果,进一步,我们证明了散射矩阵的部分信息也能局部唯一的确定位势,即反向逆散射问题的局部唯一性,并且我们改进了二阶Schr?dinger方程反向逆散射问题中位势的条件.本文总共由五章构成.第一章介绍Schr?dinger方程以及逆散射问题的物理数学背景,并介绍了二阶逆散射问题的发展历史及研究现状,然后我们给出本文的主要研究内容.第二章讨论了带一阶扰动位势的高阶Schr?dinger方程的逆散射问题,我们的方法建立在预解式的渐进展开式与散射矩阵之间的联系上.在位势具有指数衰减的条件下,我们对高阶Schr?dinger算子的预解式及预解式的导数作渐进展开,再构造Poisson算子,并利用预解式的渐进展式及其与散射矩阵之间的联系我们建立了位势与散射矩阵之间的关系式,最后我们构造高阶Schr?dinger方程的复几何光学解并带入位势与散射矩阵之间关系式,从而可以证明散射矩阵唯一确定位势.第三章考虑具有非紧支位势的二阶Schr?dinger方程的反向逆散射问题,证明了具有适当衰减条件的位势可由反向散射振幅局部唯一确定;并且对于高阶Schr?dinger方程,我们在适当条件下也证明了相应的结果.第四章考虑高阶Schr?dinger方程的Strichartz估计,推广了二阶情形的部分已知结果.第五章主要是对本论文的总结并讨论进一步可研究的内容.更多还原

【Abstract】 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.更多还原