Asymptotic and numerical analysis of resonances in circular and elliptic quantum potentials

The complex wavenumber plane endows us with a natural setting in which to appreciate and trace the behaviour of the various types of solutions of the time-independentSchr¨odinger equation in response to changes in the parameters of the potential. Whenthis is done for simple potential systems in one and two dimensions, surprisingly richdynamics emerge, such as the presence of attracting points, and interesting interconversions between the various kinds of solutions. Simple one dimensional squarepotentials, both attractive and repulsive were treated first. The calculations were thenperformed for a 2D cylindrical barrier, a cylindrical well, and the case of a slightly elliptic barrier potential. This work is a systematic study of resonance trajectories in thecomplex wavenumber plane, and an additional complex potentiodynamic plane, whichis more suited to studying the asymptotic behaviour of the states as the potential becomes infinitely high, or infinitely deep. We investigate the behaviour of bound states,external and internal resonances and antiresonances, and antibound states.Modern computational approaches to complex transcendental root finding were implemented and combined to provide us with robust and tunable tools for conductingthe numerical work. A library of Mathieu functions of complex parameters was created(as no such library exists in commercially available software at present), which gave usthe freedom to study an elliptic potential in the complex k plane.