Spectral Properties of the Anti-Periodic Boundary-Value-Transition Problems

Abstract

This work is concerned with the boundary-value-transition problem consisting of a two-interval Sturm-Liouville equation Lu ≔ −u′′(x) + q(x)u(x) = λu(x) , x ∈ [−1,0) ∪ (0,1] together with anti-periodic boundary conditions, given by u(−1) = −u(1) u′(−1) = −u′(1) and transition conditions at the interior point x = 0, given by u(+0) = Ku(−0) u′(+0) =1/Ku′(−0) where q(x) is a continuous function in the intervals [−1,0) and (0,1] with finite limit values q(±0) , K ≠ 0 is the real number and λ is the complex eigenvalue parameter. In this study we shall investigate some properties of the eigenvalues and eigenfunctions of the considered problem.