Interacting bosons in a nearly resonant potential well

We establish that the ability of a localized trapping potential to bind weakly interacting bosons is dramatically enhanced in the vicinity of the threshold of formation of the single-particle bound state of the trap. Specifically, for repulsive particles and a superthreshold trapping potential the equilibrium number of bound bosons and the size of the ground state diverge upon approaching the single-particle threshold from above. For attractive interactions and a subthreshold trap, a collective bound state always forms for a sufficiently large number of bosons, despite the inability of interparticle attraction alone to form a two-body bound state.