摘要
本文讨论了以下两个带周期边界条件的常微分算子的特征值问题。
1 周期边界条件下Sturm-Liouville问题
2 周期边界条件下四阶常微分算子特征值问题分别得到了整函数ω_1(λ)，ω_2(λ)。它们的零点集合与相应特征值问题的特征值集合重合。在此前提下，本文证明了两个特征值问题特征值的秩和其作为ω_1(λ)，ω_2(λ)零点的重数一致，从而用留数方法证明了第一个特征值问题的特征展开定理，并对特征值迹公式的合理性进行了说明。另外，用全连续算子方法证明了第二个特征值问题的特征展开定理。
The present paper will research on the rank of the eigenvalue of two
ordinary differtial operator under periodic boundary condition.
One is the Sturm-Liouville problem under periodic boundary condition
The other is a four order ordinary differtial operator under periodic boundary condition
By analysising and calculating, two entire function w1(?) and W2(?) are obtained expectively, whose zero set coincedents the set of eigenvalue of the corresponding eigenvalue problem. On this premise, the paper furthermore proves that the rank of the eigenvalue equals the order of the zero. As an application, the expansion theorem of the first eigenvalue problem was proved by resorting to the residue method, and the reasonableness of the trace identity was explained clealy. As far as the second eigenvalue problem was concerned, the expansion theorem was proved by the theory of completely continuous operator (compact operator).
引文
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