Analytical solution to the Lippmann–Schwinger equation in the spherical space

Abstract

The purpose of this work is to obtain an analytical solution to the Lippmann–Schwinger equation for a scalar particle, bounded in a two-dimensional space with constant positive curvature, called the spherical space. Making use of the thin-layer quantization method, we present the free particle wave function solving the Schrödinger equation on the spherical space using the Laplace–Beltrami operator. We use the Green’s function on the spherical space as the kernel in the Lippmann–Schwinger equation and solve it exactly for the boundary-wall potential, modeled as Dirac delta distributions, considering single, double and finding the generalized result for multiple consecutive barriers. The probability densities are presented graphically.