The analysis of scattering data is usually done by fitting the S-matrix at real experimental energies. An analytic continuation to complex and negative energies must then be performed to locate possible resonances and bound states, which correspond to poles of the S-matrix. Difficulties in the analytic continuation arise since the S-matrix is energy dependent via the momentum, k and the Sommerfeld parameter, η, which makes it multi-valued. In order to circumvent these difficulties, in this work, the S-matrix is written in a semi-analytic form in terms of the Jost matrices, which can be given as a product of known functions dependent on k and η, and unknown functions that are entire and singled-valued in energy. The unknown functions are approximated by truncated Taylor series where the expansion coefficients serve as the data-fitting parameters. The proper analytic structure of the S-matrix is thus maintained. This method is successfully tested with data generated by a model scattering potential. It is then applied to α12C scattering, where resonances of 16O in the quantum states Jρ =0+, 1−, 2+, 3−, and 4+ are located. The parameters of these resonances are accurately determined, as well as the corresponding S-matrix residues and Asymptotic Normalisation Coefficients, relevant to astrophysics. The method is also applied to dα scattering to determine the bound and resonance state parameters, corresponding S-matrix residues and Asymptotic Normalisation Coefficients of 6Li in the 1+, 2+, 3+, 2−, and 3− states. / Thesis (PhD)--University of Pretoria, 2020. / National Research Foundation (NRF) / Physics / PhD / Unrestricted

<p>  The adiabatic hyperspherical representation has been applied to study few-body systems in both ultracold atomic physics and low energy nuclear physics, as it is a powerful tool that can be used to solve a variety of few-body Hamiltonian's across a wide range of disciplines in physics. In conjunction with the adiabatic hyperspherical representation, we utilized an explicitly correlated Gaussian basis expansion, different from the traditional hyperspherical harmonic expansion typically used with this method. In atomic physics, we applied this method to study the four-body (e^-e^-e⁺e⁺) coulombic system to study positronium-positronium collisions and to get a theoretical value of the 1<em>s</em>-2<em>s</em> scattering length. This work is published in [Phys. Rev. A 100, 012711 (2019)]. We also looked at few-body physics near the unitary limit, particularly near the <em>s</em>- and <em>p</em>-wave unitary limits where the dominant length scale is the scattering length and scattering volume. On this front, we studied universal physics in this regime for the equal-mass system. This work is published in [Phys. Rev. A 106, 023304 (2022)]. This method was further applied to few-body nuclear physics.</p> <p><br></p> <p>  We treat the three and four neutron scattering problems in the <em>N</em>-body continuum to understand and gain insight into possible few-neutron resonances, most notably whether a four-neutron resonance exists. There have been many conflicting theoretical results on whether a four-neutron resonance exists that stemmed from a recent experiment by Kisimori et al. in 2016 [Phys. Rev. Lett. 116, 052501 (2016)]. To provide further theoretical insight on this problem, we use the adiabatic hyperspherical toolkit to probe the scattering continuum and from the 4<em>n</em> density of states, conclude that there is no 4<em>n</em> resonance state. Our work on this is published in [Phys. Rev. Lett. 125, 052501 (2020)] and [Phys. Rev. C 103, 024004 (2021)]. There are other few-body systems in nuclear physics that are explored in the adiabatic hyperspherical representation, including systems like the triton, helium-3, and helium-4 nuclei to visualize and characterize the different reaction pathways the <em>N</em>-body system can fragment into at a given collision energy.</p>

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