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

By assumption, a slow quantum particle moves in the two-dimensional plane of the three-dimensional configuration space and its movement takes place in the field of a central short-range potential. It is shown that the approximated energies of the weakly-bound and near-threshold resonance states of this particle are defined via the roots of the transcendental equations containing two parameters: the scattering length and effective range. The sufficient conditions for solvability of those equations are found. The dependence of their solutions on the parameters is studied.