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Zero Energy Ground State in the Three-Body System

We consider a 3--body system in $\mathbb{R}^3$ with non--positive potentials and non--negative essential spectrum. Under certain requirements on the fall off of pair potentials it is proved that if at least one pair of particles has a zero energy resonance then a square integrable zero energy ground state of three particles does not exist. This complements the analysis in \cite{1}, where it was demonstrated that square integrable zero energy ground states are possible given that in all two--body subsystems there is no negative energy bound states and no zero energy resonances. As a corollary it is proved that one can tune the coupling constants of pair potentials so that for any given $R, ε>0$: (a) the bottom of the essential spectrum is at zero; (b) there is a negative energy ground state $ψ(ξ)$, where $\int |ψ(ξ)|^2 = 1$; (c) $\int_{|ξ| \leq R} |ψ(ξ)|^2 < ε$.