Two-linked periodic Sturm-Liouville problems with transmission conditions

Abstract

The aim of this study is to generalize some important spectral properties of classical periodic Sturm-Liouville problems to two-linked periodic problems with additional transmission conditions at an interior point of interaction, which have not been studied previously. Although the classical periodic Sturm-Liouville problem has infinitely many real eigenvalues, there are two-linked periodic Sturm-Liouville problems that have only a finite number of real eigenvalues. To show this, we construct a simple example of two-linked periodic Sturm-Liouville problem with transmission conditions having only one real eigenvalue. Using our own approach, we define the characteristic determinant of a new type, find simple conditions on the coefficients of the transmission conditions that guarantee the existence of an infinite number of real eigenvalues, and derive an asymptotic formula for the eigenvalues.