Energies of weakly bound and near-threshold resonance states of a quantum particle in a two-dimensional plane

We assume that a slow quantum particle moves in a two-dimensional plane of a three-dimensional coordinate space and its motion occurs in the field of a central short-range potential. We show that the approximate energies of weakly bound and near-threshold resonance states of this particle are defined by the roots of transcendental equations with two parameters: the scattering length and the effective radius. We find the sufficient conditions for solvability of these equations and study the dependence of their solutions on the parameters.