Analytic models of ultra-cold atomic collisions at negative energies for application to confinement-induced resonances

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.