Scattering in the Gaussian Space in Semi-Classical Approximation

Abstract

Gaussian space is a three-dimensional Gaussian hypersurface embedded in a fourdimensional Euclidean space. The curvature of space depends on the distance to the origin. At a large distance, such a space is practically Euclidean, and because of this there are solutions to the Schr¨odinger equation that behave at infinity like plane waves. Without introducing any additional potential, the problem of particle’s scattering in the Gaussian space is considered in semi-classical approximation. The role of the scattering center is played by the space itself, which is strongly curved at the origin. In this paper the complete WKB (Wentzel-–Kramers Brillouin) solutions to the Schr¨odinger equation have been built. Approximate expressions for the scattering phase shifts were obtained. The total cross section energy dependence was calculated numerically using these phase shifts. Both results display the tendency of the cross section to a constant value at high energy regime.