Some properties of a periodic Sturm-Liouville problem with discontinuity conditions

In this study, some properties of the periodic Sturm-Liouville boundary-value problem given on two disjoint intervals are investigated. In order to demonstrate that the eigenvalues of the periodic Sturm-Liouville boundary-value-transition issue are real, the eigenfunctions corresponding to various eigenvalues are orthogonal, and the problem is self-adjoint, we defined a new inner product in the (L2[-π,0) ⊕ L2(0,π] ) ⊕ ℂ direct sum space. Most of eigenvalue problems require extensive consideration, analysis, and a close examination of the qualitative behavior of both the eigenvalues and the eigenfunctions since they cannot be solved explicitly. Therefore, we presented a useful method for examining the qualitative characteristics of eigenvalues and their corresponding eigenfunctions. The difference between the problem investigated here from other problems is that it is a periodic Sturm–Liouville problem that includes the parameter λ in the transition conditions.