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 those of the one- and three-dimensional cases. The energy shifts are well reproduced by a scattering length formula ?E = A20/ln?(a/R), where a is the scattering length in the short-range potential, A20/(2??) is the square of the wavefunction 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.