We consider finite-range effects when the scattering length goes to zero near a magnetically controlled
Feshbach resonance. The traditional effective-range expansion is badly behaved at this point,
and we therefore introduce an effective potential that reproduces the full T-matrix. To lowest order
the effective potential goes as momentum squared times a factor that is well defined as the scattering
length goes to zero. The potential turns out to be proportional to the background scattering
length squared times the background effective range for the resonance.
We proceed to estimate the applicability and relative importance of this potential for Bose-Einstein condensates
and for two-component Fermi gases
where the attractive nature of the effective potential can lead to collapse above a critical particle number
or induce instability toward pairing and superfluidity. For broad Feshbach resonances the higher order effect is
completely negligible. However, for narrow resonances in tightly confined samples signatures might be
experimentally accessible. This could be relevant for suboptical wavelength microstructured traps
at the interface of cold atoms and solid-state surfaces.
Effective Potential for Ultracold Atoms at the Zero Crossing of a Feshbach Resonance
