Effective-range theory in two dimensions

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Abstract
A parametrization of low‐energy two‐dimensional quantum scattering is given which is in the tradition of familiar three‐dimensional effective‐range theory. A scattering length differing from previous treatments is introduced whose behavior reflects the repulsive or attractive nature of the interaction, and whether or not bound states are present. However its definition is subject to an unknown length parameter. An effective‐range parameter is also defined and together with the scattering length explains the nonexistence of the Efimov effect in two dimensions. It is also shown that the Ramsauer–Townsend effect exists in 2D. Application to the case of a square‐well potential reveals direct relationships between the unknown length parameter and the effective range to the actual range of the potential.