  We revisit the problem of a BCS superconductor in the regime where the Fermi
energy is smaller than the Debye energy. This regime is relevant for
low-density superconductors such as SrTiO$_3$ that are not in the BEC limit, as
well as in the problem of "shape resonances" associated with the confinement of
a three-dimensional superconductor. While the problem is not new, exact results
were lacking in the low-density limit. In two dimensions, we find that the
initial rise of the pairing temperature $T_c$ at low density $n$ is nonanalytic
and faster than any power of $n$. In three dimensions, we also find that $T_c$
is nonanalytic, but starts with zero slope at weak coupling and infinite slope
at strong coupling. Self-consistent treatment of the chemical potential and
energy dependence of the density of states are crucial ingredients to obtain
these results. We also present exact results for multiband systems and confirm
our analytical expressions by numerical simulations.
