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An asymptotically exact many body theory for spin polarized interactingfermions in a one-dimensional harmonic atom trap is developed using thebosonization method and including backward scattering. In contrast to theLuttinger model, backscattering in the trap generates one-particle potentialswhich must be diagonalized simultaneously with the two-body interactions.Inclusion of backscattering becomes necessary because backscattering is thedominant interaction process between confined identical one-dimensionalfermions. The bosonization method is applied to the calculation of one-particlematrix elements at zero temperature. A detailed discussion of the validity ofthe results from bosonization is given, including a comparison with directnumerical diagonalization in fermionic Hilbert space. A model for theinteraction coefficients is developed along the lines of the Luttinger modelwith only one coupling constant $K$. With these results, particle densities,the Wigner function, and the central pair correlation function are calculatedand displayed for large fermion numbers. It is shown how interactions modifythese quantities. The anomalous dimension of the pair correlation function inthe center of the trap is also discussed and found to be in accord with theLuttinger model.Comment: 19 pages, 5 figures, journal-ref adde

Treatment of backscattering in a gas of interacting fermions confined to a one-dimensional harmonic atom trap