  A many body theory for a two-component system of spin polarized interacting
fermions in a one-dimensional harmonic trap is developed. The model considers
two different states of the same fermionic species and treats the dominant
interactions between the two using the bosonization method for forward
scattering. Asymptotically exact results for the one-particle matrix elements
at zero temperature are given. Using them, occupation probabilities of
oscillator states are discussed. Particle and momentum densities are calculated
and displayed. It is demonstrated how interactions modify all these quantities.
An asymptotic connection with Luttinger liquids is suggested. The relation of
the coupling constant of the theory to the dipole-dipole interaction is also
discussed.
