论文摘要
所谓微分算子主要研究两个方面的问题,一方面研究微分算子的谱问题,另一方面研究微分算子的逆谱问题。所谓逆谱问题就是由谱数据的信息,尤其是特征值,确定微分算子进而将其重构。本文针对Sturm-liouville问题和Dirac系统的逆谱问题进行研究,主要内容及结果如下：1、作为逆问题的预备,对Sturm-liouville问题和Dirac系统的内容进行阐述。回顾两种算子的研究历史以及特征值和特征函数的内容。进而,在两种算子下给出Weyl函数的定义,并刻画了它的性质。2、确定Sturm-liouville算子。对于Sturm-liouville算式,基于Dirichlet边值条件和非Dirichlet边值条件下,势函数满足一定的条件,同时给出一定数量的谱数据,应用Weyl函数的方法唯一确定势函数,从而确定Sturm-liouville系统。其中,势函数的条件和谱数据的数量有一定关系。3、确定Dirac系统。对于Dirac算子,在相应的边值条件下,势函数满足一定的条件,同时给出一定数量的谱数据,应用Weyl函数的方法,便能唯一确定势函数,从而确定Dirac系统。其中,势函数的条件和谱数据的数量有一定关系。本文分别对势函数多半区间已知和少半区间已知的情况做了详细的研究。
论文目录
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