Trace Formula For The System Of Equation About Sturm-liouville Eigenvalue Problem With Two Different Boundary Conditions

Letwhere u(x),w(x), v(x)∈C~2[0,π].In this paper, the traces formula of eigenvalue problem Ly=λy with two different boundaryconditions are researched. The boundary condition respectivelyFirst, we construct an entire function by use of the asymptotic estimations of solutions of initial value problem for Ly =λy .The zero set of the entire function coincide with the eigenvalue set of Ly =λy under corresponding boundary condition. By restorting to the integral identity:Using the residue method, we obtain two trace formulas respectively for above two problems.