一致预解式估计与高阶薛定谔算子

发布时间：2018-01-07 21:15

本文关键词：一致预解式估计与高阶薛定谔算子　出处：《华中科技大学》2016年博士论文　论文类型：学位论文

【摘要】：本文主要研究泛函分析与调和分析在Schrodinger算子及Schrodinger方程的某些Lp问题中的应用.首先我们将讨论Laplace算子在紧流形上的一致预解式估计,并考虑分数次Laplace算子的一致估计.此外,我们将改进高阶Schrodinger算子有关定量唯一性问题的一个结果.最后,我们考虑分数阶Schrodinger方程解的Lp估计.本篇博士论文共分为六章.第一章介绍Schrodinger方程以及一致Sobolev估计的背景及研究现状,并给出本文的研究内容.第二章讨论一致Sobolev估计在常曲率空间形式中的推广.我们的方法建立在预解算子与波动方程之间的联系上.特别地,在球面的情形,其主要创新之处在于充分利用cost(?)-△Sn+(n-1/2)2关于时间t是周期的这一特点来得到最优估计；而在负曲率的情形,我们通过将欧氏空间中著名的Stien-Tomas限制性定理推广至双曲空间中来得到相应的预解估计.第三章研究一致Sobolev估计推广至分数次Laplace算子的情形,这里,我们借助分数次算子预解式的表示,本质上转化成二阶的情形.此外,在带位势的情形下,利用Fredholm理论,我们将建立相应的极限吸收原理.第四章考虑一类高阶Schrodinger算子的定量唯一性问题，其主要想法是通过选取合适的权函数,来证明相应的Carleman估计,其结果改进和推广了部分已知结果.第五章首先建立带Kato位势的分数次Schrodinger算子所对应热核的逐点估计，在证明中我们依赖于Kato位势与分数次Laplace算子预解式的密切联系.随后,借助于热核的逐点估计,我们得到相应分数次Schrodinger方程解的Lp估计.第六章主要是对本论文的总结并讨论进一步可研究的内容.
[Abstract]:In this paper, we mainly study the application of functional analysis and harmonic analysis to some LP problems of Schrodinger operator and Schrodinger equation. Firstly, we will discuss Laplace calculation. The uniformly resolvent estimators of subs on compact manifolds. We also consider the uniform estimation of fractional Laplace operators. In addition, we will improve a result on the quantitative uniqueness of higher order Schrodinger operators. Finally. We consider the LP estimate of the solution of fractional Schrodinger equation. This doctoral thesis is divided into six chapters. Chapter 1 introduces the Schrodinger equation and uniform Sobolev estimate. Background and research status. In chapter 2, we discuss the generalization of uniform Sobolev estimator in the form of constant curvature space. Our method is based on the relation between the resolvent operator and the wave equation. In the case of sphere, the main innovation is to make full use of Costco? The optimal estimate is obtained by the characteristic that the time t is the period. And in the case of negative curvature. By extending the famous Stien-Tomas restrictive theorem in Euclidean spaces to hyperbolic spaces, we obtain the corresponding resolvent estimates. In chapter 3, we generalize the uniform Sobolev estimates to fractional ones. The case of the Laplace operator. In this paper, we use the representation of fractional operator to transform into a second order essentially. In addition, in the case of potential, we use the Fredholm theory. In Chapter 4th, we consider the quantitative uniqueness of a class of higher order Schrodinger operators. The main idea is to select appropriate weight functions. To prove the corresponding Carleman estimate. The results improve and generalize some known results. In Chapter 5th, the pointwise estimates of the thermal kernels corresponding to fractional Schrodinger operators with Kato potential are first established. In the proof, we rely on the close relation between the Kato potential and the resolvent of fractional Laplace operator. Then, with the help of the point-by-point estimation of the hot kernel. We obtain the LP estimate of the solutions of the fractional Schrodinger equation. Chapter 6th is a summary of this paper and discusses the contents that can be further studied.
【学位授予单位】：华中科技大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O177

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