带有有限个转移条件的高阶微分算子特征函数系的完备性

Completeness of Eigenfunctions of Higher-Order Differential Operator with Finite Transmission Conditions

研究一类正则的具有混合边界条件并带有有限个转移条件的高阶不连续微分算子特征值问题以及特征函数系的完备性问题.通过结合转移条件定义的新的内积,把问题转换成一个新的Hilbert空间上的对称微分算子的特征值问题.使用分段定义的微分方程的基本解,给出了满足特征方程的特征值是一个整函数的零点,证明了问题的特征值至多可数,得到特征值的充要条件.在此基础上,结合紧算子的谱理论以及逆算子的相关性质,得到了Green函数,证明了特征函数系是完备的.

In this paper, we focus on a class of regular higher-order differential operators with finite transmission conditions and coupled boundary conditions. The eigenvalue prob- lems and the completeness of eigenfunctions are investigated. First, follwing a combination of a new inner product relating the transmission conditions, we transform the eigenvalue problems of the original operator into those of the symmetric differential operator in the new Hilbert space. Then, applying piece-wise fundamental solutions of the differential equation, we clarify that eigenvalues satisfying the eigenequation coincide with the zeros of the entire function. Moreover, on this problem, the existence of countably many eigenvalues is proved as well as the necessary and sufficient conditions of eigenvalues are established. Furthmore, by the spectral theorem of compact operator and properties of inverse operator, Green＇s function is constructed and the completeness of eigenfunctions is demostrated.