Coupling constants for a degenerate Fermi gas confined to a quasi one-dimensional harmonic trap

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A theory for the coupling constants between spin polarized degenerate fermionic atoms is developed for the case of atoms confined to a quasi one-dimensional harmonic trap. Exploiting the presence of the Fermi edge and the large number of fermions it is shown that the resulting effective one-dimensional interaction can be parameterized by three sets of interaction coefficients, two for forward scattering and one for backward scattering. In the case of identical fermions, backscattering dominates because the contact part of the effective one-dimensional interaction must be subtracted. Analytic expressions for the interaction coefficients for the effective contact interaction, which is relevant for the inter-component interactions, and for the "p-wave" interaction appearing in next order, are given. As an example, we calculate and discuss in detail the effective coupling constants for the marginally long ranged dipole--dipole interaction.