Scattering from singular potentials

In this thesis, some well-known mathematical methods are applied to a new set of problems as some bound state and scattering properties of singular potentials are investigated quantum mechanically. Firstly, we work on the hybrid singular potential systems consisting of circular and spherical delta potentials with additional pointwise delta potentials which are outside of these circle and sphere in two and three dimensional spaces. We also investigate the cases when there is a small deformation in these shells in the normal direction. We apply the cut-off regularization method to regularize the ill-defined Hamiltonian and perform a convenient renormalization process for the singularity occurring in the self-interaction terms of the point defects. We show that the first order correction to the bound state energies has a simple geometric interpretation when the circle and the sphere that support the delta potentials are slightly deformed. Secondly, we give a detailed study of scattering from linear Diracdelta potentials in two dimensions which are supported on infinitely long parallel lines normal to the scattering axis. We also explore how the presence of these lines affect the geometric scattering. We take the asymptotically flat Gaussian bump as a perturbation to the line defects system and we calculate the scattering amplitude. We give plots for one or two line defects and show that the presence of line defects, especially when the bump is located between the defects, amplifies the geometric scattering effects. We give a brief expression for the scattering amplitude for the case combining a finite size linear defect and a point defect. Lastly, we introduce a distributional direct method which is performed in momentum space to perform the bound and scattering state analysis of the spherical and circular delta potentials. The results are compatible with the standard partial wave approach.