Some Questions About 2×2 Sturm-liouville Operator

Let L =(?) , where real function u , v , w∈C~2[0,π]. The rank of the eigenvalue of 2×2 Sturm-Liouville ordinary differential operator underperiodic boundary condition and the expansion theorem under self-adjoint boundary conditionwill be researched on this paper .One is the Sturm-Liouville problem under periodic boundary conditionThe other is 2×2 Sturm-Liouville ordinary differential operator under one self-adjoint boundaryconditionBy analyzing and calculating, the two problems'entire functionω(λ) are obtained respectively, whose zero set coincides with the set of the corresponding eigenvalue problem. For thefirst problem, it is proved that the rank of the eigenvalue equals the order of the zero. As far asthe second eigenvalue problem was concerned, the expansion theorem was proved by the theoryof completely continuous operator (compact operator).