The scattering problem on two half-lines with transfer-matrix condition at the origin

Abstract

Abstract The mathematical analysis of scattering theory has been a major area of interest in mathematics and physics research since the latter half of the twentieth century. The aim of this work is to examine, in a functional analytic setting, properties of the differential operator L and solutions involved for scattering on the line −∞ < x < ∞. The characterisation of the spectrum of L will provide insight into the physical interpretation of the problem. The study of scattering theory will proceed with the major results in the field being presented with particular focus on reflectionless scattering. Attention is then directed to the inverse reflectionless case. We look at scattering on the line with a matrix transfer condition at the origin in addition an overview of the inverse case is presented.