  We consider a three-particle system in $\mathbb{R}^3$ with non-positive
pair-potentials and non-negative essential spectrum. Under certain restrictions
on potentials it is proved that the eigenvalues are absorbed at zero energy
threshold given that there is no negative energy bound states and zero energy
resonances in particle pairs. It is shown that the condition on the absence of
zero energy resonances in particle pairs is essential. Namely, we prove 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. It is
also proved that one can tune the coupling constants of pair potentials so that
for any given $R, \epsilon >0$: (a) the bottom of the essential spectrum is at
zero; (b) there is a negative energy ground state $\psi(\xi)$ such that $\int
|\psi(\xi)|^2 d^6 \xi= 1$ and $\int_{|\xi| \leq R} |\psi(\xi)|^2 d^6 \xi <
\epsilon$.
