The main question that has been answered in this research project is: what happens to the singularities of the S-matrix when the interaction potential becomes complex? In other words, we looked at the resonance spectral points and traced their movements on the complex momentum plane, when the imaginary part of the potential is gradually increased from zero to a certain reasonable value. This question is important in many fields of research where optical potentials are used. An optical potential can effectively take into account the loss of the beam of particles into all the reaction channels that are formally ignored. Such a simplified approach is widely used in nuclear and atomic physics. In order to study the movement of the spectral points, we used a simple potential and the Jost function method that allowed us to easily locate the spectral points as complex zeros of the Jost function. / Dissertation (MSc)--University of Pretoria, 2016. / Physics / MSc / Unrestricted

Recently, was developed a new theory of the Jost function, within which, it was split in two terms involving on one side, singlevalued analytic functions of the energy, and on the other, factors responsible for the existence of the branching-points. For the single-valued part of the Jost function, a procedure for the powerseries expansion around an arbitrary point on the energy plane was suggested. However, this theory lacks a rigorous proof that these parts are entire functions of the energy. It also gives an intuitive (not rigorous) derivation of the domain where they are entire. In the present study, we ll this gap by using a method derived from the method of successive approximations. / Dissertation (MSc)--University of Pretoria, 2013. / gm2014 / Physics / unrestricted

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: The tools of perturbative Quantum Field Theory are by now a standard part of the theoretical physicist's arsenal. In this thesis we investigate the spectral method, an approach that uses tools from the quantum theory of scattering to calculate the O(~) corrections to fields. Speci cally, we investigate whether the approach can be extended to deal with two interacting scalar elds with a mass gap in (1+1) dimensions. To achieve this we need to verify the analyticity of the appropriate Jost function. All the machinery to do so is introduced during the course of the thesis. This includes the eld theoretic formalism which describes such a system, and the derivation of a number of di erential equations from which the density of states can be constructed. The numerical method is also outlined. Concrete results are presented to verify that the approach reproduces known results. Arguments related to Levinson's theorem are then presented that suggest that the Jost function is indeed analytic, with some caveats. / AFRIKAANSE OPSOMMING: Die gereedskap van perturbatiewe Kwantumveld-teorie is teen di e tyd al `n standaard-deel van die teoretiese sikus se werkskis. In hierdie tesis ondersoek ons die spektrale metode, `n benadering wat gereedskap van die kwantumteorie van verstrooing gebruik om die O(~) korreksie van velde te bereken. Ons ondersoek spesi ek of die metode uitgebrei kan word om toepaslik te wees in die geval van twee wisselwerkende skalaarvelde met `n massa-gaping in (1+1) dimensies. Vir hierdie doel moet ons bepaal of die toepaslike Jost-funksie analities is of nie. Al die masjinerie om dit te bepaal word deur die loop van hierdie tesis ingevoer. Dit sluit in die veld-teorietiese formalisme wat so `n stelsel beskryf, en die a eiding van `n aantal di erensiaal vergelykings wat gebruik kan word om die digtheid van toestande te konstrueer. Die numeriese metode word ook beskryf. Konkrete resultate wat bevestig dat die metode die korrekte antwoorde in `n analities bekende geval weergee, word verskaf. Argumente wat verband hou met Levinson se stelling word gebruik om te bevestig dat die Jost-funksie inderdaad analities is, met sekere voorbehoude.

The aim of this research is to develop a method for expanding the Jost functions as a Taylor-type power series on the complex angular momentum plane. From this method in conjunction with the Watson transformation, we were able to express the scattering amplitude as a sum of the background and pole terms, furthermore, this method propose a way of evaluating, numerically, the pole term. To demonstrate how this method may be applied, we considered the Born approximation. We found out that the developed method improved the Born approximation at large scattering angles. Therefore, this method is useful when the di fferential cross section of the background term fails to converge to that of the exact diff erential cross section at large scattering angles. / Dissertation (MSc)--University of Pretoria, 2016. / National Research Foundation (NRF) / Physics / MSc

The double hypernuclear systems are studied within the context of the hyperspherical approach. Possible bound states of these systems are sought as zeros of the corresponding three-body Jost function in the complex energy plane. Hypercentral potentials for the system are constructed from known potentials in order to determine bound states of the system. Calculated binding energies for double- hypernuclei having A = 4 − 20, are presented. / Physics / M.Sc. (Physics)

The double hypernuclear systems are studied within the context of the hyperspherical approach. Possible bound states of these systems are sought as zeros of the corresponding three-body Jost function in the complex energy plane. Hypercentral potentials for the system are constructed from known potentials in order to determine bound states of the system. Calculated binding energies for double- hypernuclei having A = 4 − 20, are presented. / Physics / M.Sc. (Physics)