Inverse and Expansion Problems with Boundary Conditions Rationally Dependent on the Eigenparameter

In this paper, a boundary value problem consisting of a second-order differential equation of Sturm–Liouville type with boundary conditions rationally dependent on the eigenparameter in a bounded interval is investigated. We consider the inverse problem: given eigenvalues and eigenfunctions, we seek the corresponding potential function, uniquely. Moreover, we give the operator theoretic formula, construct the resolvent operator, and obtain the expansion formula with respect to the eigenfunctions.