Potential Scattering on a Spherical Surface

TL;DR

This paper investigates quantum potential scattering on a spherical surface, revealing how curvature and topology influence bound states and phase shifts, with implications for cold atom experiments on curved geometries.

Contribution

It provides a theoretical analysis of potential scattering on a sphere, highlighting differences from flat surfaces and deriving relations for bound state energies and phase shifts.

Findings

Bound state energies on a sphere differ from those on a plane due to curvature.

Phase shifts at energies near the sphere's scale differ significantly from flat surfaces.

Zero-energy phase shift approaches a constant related to the sphere's size, unlike the planar case.

Abstract

The advances in cold atom experiments have allowed construction of confining traps in the form of curved surfaces. This opens up the possibility of studying quantum gases in curved manifolds. On closed surfaces, many fundamental processes are affected by the local and global properties, i.e. the curvature and the topology of the surface. In this paper, we study the problem of potential scattering on a spherical surface and discuss its difference with that on a 2D plane. For bound states with angular momentum $m$, their energies ($E_{m}$) on a sphere are related to those on a 2D plane ($-|E_{m,o}|$) as $E_{m}= - |E_{m, o}| + E_{R}^{} \left[ \frac{m^2-1}{3} + O\left( \frac{r_o^2}{R^2} \right) \right] $, where $E_{R}^{} = \hbar^2/(2M R^2)$, and $R$ is the radius of the sphere. Due to the finite extent of the manifold, the phase shifts on a sphere at energies $E\sim E_{R}^{}$ differ significantly from those on a 2D plane. As energy $E$ approaches zero, the phase shift in the planar case approaches $0$, whereas in the spherical case it reaches a constant that connects the microscopic length scale to the largest length scale $R$.