Rearch On The Rank Of The Eigenvalue Of Two Ordinary Differtial Operator Under Periodic Boundary Condition

The present paper will research on the rank of the eigenvalue of twoordinary differtial operator under periodic boundary condition.One is the Sturm-Liouville problem under periodic boundary conditionThe other is a four order ordinary differtial operator under periodic boundary conditionBy 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).