  An asymptotically exact many body theory for spin polarized interacting
fermions in a one-dimensional harmonic atom trap is developed using the
bosonization method and including backward scattering. In contrast to the
Luttinger model, backscattering in the trap generates one-particle potentials
which must be diagonalized simultaneously with the two-body interactions.
Inclusion of backscattering becomes necessary because backscattering is the
dominant interaction process between confined identical one-dimensional
fermions. The bosonization method is applied to the calculation of one-particle
matrix elements at zero temperature. A detailed discussion of the validity of
the results from bosonization is given, including a comparison with direct
numerical diagonalization in fermionic Hilbert space. A model for the
interaction coefficients is developed along the lines of the Luttinger model
with only one coupling constant $K$. With these results, particle densities,
the Wigner function, and the central pair correlation function are calculated
and displayed for large fermion numbers. It is shown how interactions modify
these quantities. The anomalous dimension of the pair correlation function in
the center of the trap is also discussed and found to be in accord with the
Luttinger model.
