Phase-space organizations in prolate and oblate potentials: Classical, semiclassical, and quantum results.

Abstract

The three-dimensional trajectories of a neutron in an average deformed potential are studied for two different potentials with prolate or oblate ellipsoidal deformations. The simplest one is a simple cavity which is shown to be integrable. The motion is studied and is found to be separable in spheroidal coordinates. It is shown that the phase space of an oblate cavity may contain a separatrix which is associated with the crossing of the focal circle while the prolate phase space does contain a separatrix only for the planar trajectories. By using arguments based on a uniform semiclassical approximation, it is shown that the quantum-energy-level spectra present indeed a difference between prolate and oblate shapes corresponding to the crossing of this separatrix. The second potential studied, the Buck-Pilt potential, has a diffuse surface. When it is deformed it becomes unintegrable. For this potential Poincar\'e surfaces of section are drawn. Large regular regions are shown in which the organization of the phase space is that of the cavity. Chaos and nonlinearities are also seen. Consequences of the existence of the regular regions similar to the cavity are drawn for the energy spectrum.