Spectral Analysis of α-Semi Periodic 2-Interval Sturm-Liouville Problems

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 α > 0 . We will call this problem α -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 [ - π , π ] when the parameter α ≠ 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 α = 1 our problem is transformed into classical periodic Sturm-Liouville problems.