Binding three or four bosons without bound subsystems

Abstract

We estimate the ratio ${R=g}_{3}{/g}_{2}$ of the critical coupling constants ${g}_{2}$ and ${g}_{3}$ that are required to achieve binding of two or three bosons, respectively, with a short-range interaction, and examine how this ratio depends on the shape of the potential. Simple monotonous potentials give $R\ensuremath{\simeq}0.8.$ A wide repulsive core pushes this ratio close to $R=1.$ On the other hand, for an attractive well protected by an external repulsive barrier, the ratio approaches the rigorous lower bound $R=2/3.$ We also present results for $N=4$ bosons, sketch the extension to $N>4,$ and discuss various consequences.