Spectral Analysis of $\alpha$-Semi Periodic 2-Interval Sturm-Liouville Problems

Abstract

In this paper, we propose a new type of boundary value problems for 2-interval Sturm-Liouville equations which differs from the classical periodic Sturm-Liouville problems in that, the boundary and transmission conditions depend on a positive parameter $\alpha >0.$ We will call this problem $\alpha$-semi periodic Sturm-Liouville problem. It is important to note that our problem is not self-adjoint in the classical Hilbert space of square-integrable functions $L_2[-\pi ,\pi ]$ when the parameter $\alpha \ne 1$. First by using an our own approach we investigated some properties of eigenvalues and their corresponding eigenfunctions. Then, for self-adjoint realization of the problem under consideration we define a different inner product in the classical Hilbert space in which we treated an operator-theoretic formulation. The results obtained generalize and extend similar results of the classical periodic Sturm-Liouville theory, since in the special case $\alpha =1$ our problem is transformed into classical periodic Sturm-Liouville problems.