  We construct simple analytic models of the $S$-matrix, accounting for both
scattering resonances and smooth background contributions for collisions that
occur below the s-wave threshold. Such models are important for studying
confinement-induced resonances such as those occurring in cold collisions of
$^{133}$Cs atoms in separated sites of a polarization-gradient optical lattice.
Because these resonances occur at negative energy with respect to the s-wave
threshold, they cannot be studied easily using direct numerical solutions of
the Schr\"{o}dinger equation. Using our analytic model, we extend previous
studies of negative-energy scattering to the multichannel case, accounting for
the interplay of Feshbach resonances, large background scattering lengths, and
inelastic processes.
