Advanced Mathematical Models & Applications

In this paper, we examine a quadratic pencil for the Sturm-Liouville equation with the discontinuous coefficient. This equation arises in quantum scattering theory as part of solving an inverse problem related to the massless Klein-Gordon equation with a static potential . Many other physical scattering scenarios also reduce to this equation for instance, problems in absorbing media within transmission line theory, electromagnetism, and elasticity. We construct integral representations for the solution satisfying the Jost condition at infinity and investigate the properties, related with the coefficients of the considered equation. We also study the solution of the initial value problem wich has an important role in the investigation of the inverse scattering problem for the quadratic pencil of the Sturm-Liouville equation. Assuming a potential function belongs to class of bounded functions we give a new integral representation for the Jost solution of and prove some interesting properties for the kernel function wich can be applied for the solution of the inverse scattering problem.