Coupled-channel pseudo-potential description of Feshbach resonance in two dimensions

We derive pseudo-potentials that describe the scattering between two particles in two spatial dimensions for any partial wave $m$, whose scattering strength is parameterized in terms of the phase shift $\delta_{m}$. Using our $m=0$ pseudo-potential, we develop a coupled channel model with 2D zero-range interactions, which describes the two-body physics across a Feshbach resonance. Our model predicts the scattering length, the binding energy and the ``closed channel molecular fraction'' of two particles; these observables can be measured in experiments on ultracold quasi-2D atomic Bose and Fermi gases with present-day technology.