Impulsive spectral problems for multi-interval differential equations with periodic boundary conditions

Abstract Our motivation for this work comes from the Earth’s free oscillations, in which the relationship determining the radial dependence of its toroidal oscillations leads to the discontinuous Sturm-Liouville problem (SLP) subject to the periodic boundary conditions and additional impulsive conditions. Discontinuities in the elastic parameters lead to a discontinuity of the potential, as well as the discontinuity of the eigenfunctions. The impulsive conditions represent the displacement and stress at the interface. It should be emphasized that not all properties of the classical Sturm-Liouville problems remain valid for similar problems with discontinuities. The main difficulty lies in the character of eigenvalues and eigenfunctions. For example, although classical regular SLPs have infinitely many real eigenvalues, similar problems with discontinuities may not have infinitely many real eigenvalues and in some cases may have no real eigenvalues. In this work, we shall consider a discontinuous Sturm-Liouville problem subject to periodic boundary conditions and additional impulsive conditions. The impulsive conditions are posed at two interior points of discontinuity. Naturally, the eigenfunctions for the considered problem may be discontinuous at these points. We extend and generalize some methods and results of classical Sturm-Liouville theory to construct the fundamental solutions and derive the asymptotic formulas for the eigenvalues. We also construct an example of a periodic Sturm-Liouville problem with impulsive conditions that does not have infinitely many real eigenvalues.