Inverse problem for Hill equation with jump conditions

Abstract

In the present study, the inverse nodal problem is discussed for the discontinuous periodic (or anti-periodic) Sturm-Liouville problem. Such types of problems are different from regular problems because of discontinuity in the boundary conditions. Firstly, results of the Sturm-Liouville problem including jump condition are present. Then, by deferring the zeros of eigenfunctions, the inverse problem is solved as we desired. The method is based on considering a translation so that the periodic (or anti-periodic) problem is reduced to a Dirichlet problem as in Cheng and Law (Inverse Prob 22: 891-901, 2006). But, our problem including also discontinuous conditions. In addition to all of these, although there are so many results on this subject, we combine both periodic conditions and discontinuity.