【作者】 李慧；

【导师】 魏广生；

【作者基本信息】 陕西师范大学 ， 计算数学， 2016， 硕士

【摘要】 Dirac微分算子的逆谱问题起源于对广义量子学中自由电子变化规律的探究.随着数学物理、地球物理学、系统科学等交叉学科的发展,Dirac算子的逆谱问题已逐渐成为应用数学的一个热门研究领域.本文研究在不同谱信息已知的情况下,如何唯一确定并重构势函数的问题,主要分为两类情形：一类是常型Dirac算子的逆谱问题,另一类是边值条件含谱参数的Dirac算子的逆谱问题.主要内容安排如下：第一章系统介绍了Dirac算子逆谱问题的研究背景、意义及现状,给出了Dirac算子谱理论的相关概念及结论,为逆谱问题的研究奠定基础.第二章研究边值条件含谱参数的Dirac算子的逆三组谱问题.给出了在边值条件含谱参数的情况下,三组谱确定势函数及边值条件的唯一性定理.第三章研究常型Dirac算子的半逆谱问题.当势函数对(p(x),q(x))在一半区间上已知时,利用插值公式给出了一组谱唯一确定势函数的重构算法.第四章研究边值条件含谱参数的Dirac算子混合谱数据的逆谱问题.证明由势函数的部分信息以及部分特征值确定整个区间上势函数的唯一性定理.更多还原

【Abstract】 The inverse spectral problem of Dirac differential operator originated in re-searching the variation of free electrons in generalized quantum theory. As the development of cross subject such as mathematical physics, geophysics and system science, it has gotten more and more attention and became one of popular topic in applied mathematics. In this paper, we research how to determine and recover a po-tential in different spectral information. Here, it can be divided into two categories: one is regular boundary-value problem, another is problem with spectral parameter contained in the boundary conditions. The main works are given as follows:In the first chapter, we introduce the research background, significance, and advance of Dirac system, and review the basic concept of Dirac problem, which make preparation for the inverse spectral problem.In the second chapter, we are concerned with the inverse three spectral problem with spectral parameter in the boundary conditions. The uniqueness theorem and its proof are given.In the third chapter, the half inverse spectral problem of regular Dirac operator is considered. We obtain that the whole potential can be uniquely determined by a group of Dirichlet specturm when the potential in half of the interval are known.In the last chapter, the inverse spectral problem based on the blend spectral data, and with spectral parameter in the boundary conditions are considered. When potential in half of the interval are known, the potential can be uniquely determined.更多还原