带有三个转移条件的Sturm-Liouville有限谱问题及其矩阵表示

Sturm-Liouville finite spectrum problem with three transmission conditions and its matrix representation

主要研究带有三个转移条件的Sturm-Liouville有限谱问题.首先通过构造一类正则的带有三个转移条件的Sturm-Liouville问题,验证其恰有nl个特征值,进而表明带有三个转移条件的Sturm-Liouville问题等价于一类矩阵特征值问题,且其具有相同的特征值.此外,证明了这nl个特征值在非自共轭边界条件下可位于复平面内任何位置,在自共轭边界条件下可位于实轴上任何位置的结论.分析的关键是判断函数的迭代,运用的主要工具是Rouche定理.

In this paper,we study the Sturm-Liouville finite spectrum problem with three transmission conditions.Firstly,we construct a class of regular Sturm-Liouville problems with three transmission conditions,and verify that it has exactly nl eigenvalues.Furthermore,we show that this kind of Sturm-Liouville problems with three transmission conditions is equivalent to a class of matrix eigenvalue problems in the sense that they have exactly the same eigenvalues.In addition,we show that these nl eigenvalues can be distributed arbitrarily throughout the complex plane in the non-self-adjoint case and anywhere along the real line in the self-adjoint case.The key to this analysis is an iterative construction of the characteristic function,the main tool used in this paper is Rouche theorem.