Length and effective range of two-dimensional scattering of a quantum particle by a central short-range potential

The derivation and qualitative analysis of the nonlinear and linear equations adopted for calculation of the scattering length and effective range are given. In the case of the central square-well potential, the exact solutions of these equations are found and studied. The connection between the effective range and scattering length is revealed. Special attention is paid to the cases of zero and unrestricted scattering lengths.