Complex Distortion of Energy Spectrum in Computable Representations of Scattering Amplitudes.

Complex rotation and L^2 -discretization methods applied to the calculation of t -matrix elements between continuum functions have been shown by Nuttall and coworkers to be divergent with respect to increasing basis size. This thesis gives two generalizations of complex rotation of the coordinates of the Hamiltonian to deal with the problem. The first generalization results in a complex-valued Hamiltonian which is expressed in terms of a Sturm-Liouville operator. This approach gives semisatisfactory results in numerical tests reported here. The second generalization is to perform complex rotation or complex contour distortions in momentum space and by introducing an "analytic regulator," a completely satisfactory and practical result is achieved. In addition to these results, this thesis also presents an effective Hamiltonian method for the calculation of t-matrix elements and establishes its convergence properties. A proof that shows that this method is equivalent to the complex Kohn method recently discussed by other workers is also included. Finally, an entirely new approach of applying group theoretical methods to the analytical calculation of bound state energies and resonance energies of quantum mechanical systems is given. This method is capable of generating a number of potentials which have explicitly calculable spectra which have not been previously known. Two such examples are discussed.