  We apply the general principles of effective field theories to the
construction of effective interactions suitable for few- and many-body
calculations in a no-core shell model framework. We calculate the spectrum of
systems with three and four two-component fermions in a harmonic trap. In the
unitary limit, we find that three-particle results are within 10% of known
semi-analytical values even in small model spaces. The method is very general,
and can be readily extended to other regimes, more particles, different species
(e.g., protons and neutrons in nuclear physics), or more-component fermions (as
well as bosons). As an illustration, we present calculations of the
lowest-energy three-fermion states away from the unitary limit and find a
possible inversion of parity in the ground state in the limit of trap size
large compared to the scattering length. Furthermore, we investigate the lowest
positive-parity states for four fermions, although we are limited by the
dimensions we can currently handle in this case.
