Simplified contact pseudopotential for anisotropic interactions of polarized particles under harmonic confinement

Motivated by the recent progress in cooling and trapping polar molecules, we present a simplified version of the rigorous contact pseudopotential for anisotropically-interacting polarized particles [A. Derevianko, Phys. Rev. A 67, 033607 (2003); Phys. Rev. A 72, 039901(E) (2005)]. The simplifications are carried out for a practically important case of harmonically confined particles described by sufficiently smooth wavefunctions. The resulting contact pseudo-potential depends on the K-matrix of the underlying scattering process and is represented as a sum over pairs of partial waves coupled by the collision. The contribution of each pair of the coupled waves (l and l') involves a tensor product of derivatives of orders l and l'. The asymmetry in the derivatives reflects the anisotropy of the original interaction potential: there is a preferential appearance of the derivatives along the polarizing field.