Pseudo-potential of a power-law decaying interaction in two-dimensional systems

Abstract

We analytically derive the general pseudo-potential operator of an arbitrary isotropic interaction for particles confined in two-dimensional (2D) systems, using the frame work developed by Huang and Yang for 3D scattering. We also analytically derive the low energy dependence of the scattering phase-shift for an arbitrary interaction with a power-law decaying tail, $V_{2D}(\rho )\propto {\rho }^{-\alpha }$ (for $\alpha >2$). We apply our results to the 2D dipolar gases ($\alpha =3$) as an example, calculating the momentum and dipole moment dependence of the pseudo-potential for both $s$- and p-wave scattering channels if the two scattering particles are in the same 2D layer. Results for the s-wave scattering between particles in two different (parallel) layers are also investigated. Our results can be directly applied to the systems of dipolar atoms and/or polar molecules in a general 2D geometry.

4 pages and 2 figures. Some correction is made for the pseudo-potential of higher angular momentum. Mre references are added

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