Effective-radius approximation in the problem of two-dimensional scattering by a central short-range potential

We study the role of weakly bound and near-threshold resonance states of a quantum particle in its two-dimensional elastic low-energy scattering. The energies of such states are determined in the effective-radius approximation using the roots of transcendental equations. The same approximation is used to analyze scattering. We obtain and study the explicit low-energy asymptotic forms of all partial phases and cross sections. These asymptotic forms contain energies of weakly bound or near-threshold resonance states.