Exotic atoms in two dimensions

TL;DR

This paper investigates the energy level behavior of exotic atoms in two dimensions, analyzing how short-range interactions affect energy shifts and comparing results across different spatial dimensions.

Contribution

It introduces a scattering length formula to accurately predict energy shifts in two-dimensional exotic atoms, extending understanding beyond 1D and 3D cases.

Findings

Energy shifts are well described by the scattering length formula

The formula relates energy shifts to the wave function at the origin and a parameter R

Comparison with 1D and 3D cases highlights unique 2D behaviors

Abstract

We study the behavior of energy levels in two dimensions for exotic atoms, i.e., when a long-range attractive potential is supplemented by a short-range interaction, and compare the results with these of the one- and three-dimensional cases. The energy shifts are well reproduced by a scattering length formula $ \delta{E}= A_0^2/\ln (a/R)$, where $a$ is the scattering length in the short-range potential, $A_0^2/(2\,\pi)$ the square of the wave function at the origin in the external potential, and $R$ is related to the derivative with respect to the energy of the solution that is regular at large distances.