Generalized Fourier Series as Green's Function Expansion for Multi-interval Sturm-Liouville Systems

This study aims to investigate a class of boundary-value transmission problems consisting of Sturm-Liouville equation on finite number disjoint intervals together with eigenparameter-dependent boundary conditions and supplementary transmission conditions at the interior transmittal points. We introduce new Hilbert spaces for self-adjoint realization of the problem, and state the main spectral properties of eigen-values and eigenfunctions of the considered problem. Then by suggesting own approaches, we have presented a formula for Green's function and resolvent operator. Finally, we find the resolvent function for corresponding inhomogeneous problem and establish completeness relation of eigenfunctions as generalized Fourier series.