  We study a trapped system of fermions with a zero-range two-body interaction
using the shell-model Monte Carlo method, providing {\em ab initio} results for
the low particle number limit where mean-field theory is not applicable. We
present results for the $N$-body energies as function of interaction strength,
particle number, and temperature. The subtle question of renormalization in a
finite model space is addressed and the convergence of our method and its
applicability across the BCS-BEC crossover is discussed. Our findings indicate
that very good quantitative results can be obtained on the BCS side, whereas at
unitarity and in the BEC regime the convergence is less clear. Comparison to
N=2 analytics at zero and finite temperature, and to other calculations in the
literature for $N>2$ show very good agreement.
