一类不连续四阶微分算子特征函数系的完备性

Completeness of Eigenfunctions of the Discontinuous Fourth-Order Differential Operator

研究了边界条件含有特征参数的不连续四阶微分算子L的特征函数系的完备性问题.首先,在一个适当的Hilbert空间H中定义一个与问题相关的新算子T,使得T与L有相同的特征值;然后,在已知算子自共轭的基础上,结合其附带的转移条件和边界条件,得到了特征值的判别函数.利用泛函分析方法,得到自共轭算子T的特征值是下方有界的且仅有点谱,再结合紧算子的谱理论,在新空间H中,证明了算子T的特征函数系是完备的,其中算子T的特征函数系是由算子L的特征函数系扩张而成的.

The present article intends to investigate the completeness of eigenfunctions of the discontinuous operator L with eigenparameter-dependent boundary condition. Firstly, we define a linear operator T in a suitable Hilbert space H to make the eigenvalues of the operator T and L same. As the basic of self-adjointness operator T, we obtain the identifying function of eigenvalues combining with boundary conditions and transmission conditions. Then by the method of functional analysis, we know that the eigenvalues of self-adjoint operator T are bounded below and has only point-spectrum. In combination with the spectral theorem for compact operator, we prove the eigenfunctions of T are completed on H, which augmented by eigenfunctions of the operator L.