A Marchenko equation for complex interactions with a regular analytic continuation

Abstract

Using Marchenko's own method, it is shown that three elements are required for the existence of a Marchenko fundamental equation. These are a convergent sum over the discrete spectrum, a bounded translation operator, and sometimes when there are “spectral singularities,” a domain in the complex plane of the momentum k where the representation of the regular solution as a linear combination of the two Jost solutions is meaningful. Meanwhile, we prove that for a class of complex potentials that will be called regular, a variant of Marchenko's equation exists. Clarification of the relationship between the completeness of the two sets of solutions for the unperturbed and the perturbed equation on one hand and the existence of a fundamental equation on the other hand is also achieved.