The Modified Parseval Equality of Sturm-Liouville Problems with Transmission Conditions

Abstract

We consider the Sturm-Liouville (S-L) problems with very general transmission conditions on a finite interval. Firstly, we obtain the sufficient and necessary condition for $\lambda$ being an eigenvalue of the S-L problems by constructing the fundamental solutions of the problems and prove that the eigenvalues of the S-L problems are bounded below and are countably infinite. Furthermore, the asymptotic formulas of the eigenvalues and eigenfunctions of the S-L problems are obtained. Finally, we derive the eigenfunction expansion for Green's function of the S-L problems with transmission conditions and establish the modified Parseval equality in the associated Hilbert space.