Exotic atoms in two dimensions

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 $\delta E=A_0^2\mathrm{/}\ln (a\mathrm{/}R)$, where $a$ is the scattering length in the short-range potential, $A_0^2\mathrm{/}(2\pi )$ 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.