Droplets of Bosons at a Narrow Resonance

We consider bosons interacting through a narrow s-wave resonance. Such a resonance is characterized by an infinite scattering length and a large and negative effective range r₀. We argue that any number N≥3 of bosons can form a self-bound cluster with the binding energy per particle increasing as N² for 1 N (-r₀/abg)1/2, where abg is the background scattering length (between atoms and molecules). In the opposite limit N (-r₀/abg)1/2, bosons form droplets with binding energy per particle saturating to a constant value independent of the particle number. The stability of clusters and droplets when the interaction is detuned from the resonance is also studied.