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 $r_0$. We argue that any number $N\ge 3$ of bosons can form a self-bound cluster with the binding energy per particle increasing as $N^2$ for $1\ll N\ll (-r_0\mathrm{/}{a}_{\text{bg}}{)}^{1\mathrm{/}2}$, where $a_{\text{bg}}$ is the background scattering length (between atoms and molecules). In the opposite limit $N\gg (-r_0\mathrm{/}{a}_{\text{bg}}{)}^{1\mathrm{/}2}$, 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.