Near-threshold scattering, quantum reflection, and quantization in two dimensions

We study the near-threshold behavior of scattering phase shifts, quantum reflection amplitudes, and quantization functions for systems described by a circularly symmetric potential in two spatial dimensions. In contrast to the three-dimensional case, the centrifugal potential (m(2)-14)h(2)/(2Mr(2)) is nonvanishing and even attractive for s waves, m=0, and the leading near-threshold energy dependence of phase shifts and amplitudes for scattering and quantum reflection is logarithmic in this case. The near-threshold behavior of the s-wave phase shifts and amplitudes can nevertheless be characterized by a well-defined scattering length (for potentials falling off faster than 1/r(2)) and an effective range (for potentials falling off faster than 1/r(4)). For a potential with a bound state at energy E-n=-h(2)kappa(2)(n)/(2M) very near threshold, the scattering length obeys a similar to(kappa)(n)-> 02e(E)(-gamma)/kappa(n)+O(kappa(n)), with no term of order kappa(0)(n)-in contrast to the three-dimensional case. Analytical results are derived for homogeneous potentials, and the necessary modification of the effective-range expansion is given for potentials proportional to 1/r(4). For m not equal 0 we give analytical expressions for the near-threshold behavior of phase shifts as well as scattering and quantum reflection amplitudes, which are generally valid, even when m is neither integer nor half-integer.