Finite Spectrum of Sturm–Liouville Problems with Transmission Conditions Dependent on the Spectral Parameter

Abstract In this paper, we mainly study the finite spectrum of Sturm–Liouville problems with transmission conditions dependent on the spectral parameter. By analyzing on the characteristic function, we prove that this kind of Sturm–Liouville problems consist of finite number of eigenvalue and these finite eigenvalues can be located anywhere in the complex plane. It is illustrated that the number of eigenvalues not only depends on the partition of the domain interval, but also depends on the transmission conditions dependent on the spectral parameter and the boundary conditions.