Rigorous conditions for the existence of bound states at the threshold in the two-particle case

In the framework of non-relativistic quantum mechanics and with the help of Green's functions formalism, we study the behaviour of weakly bound states in a non-central two-particle potential as they approach the continuum threshold. Through estimating Green's function for positive potentials we derive rigorously the upper bound on the wavefunction, which helps us to control its falloff. In particular, we prove that for potentials whose repulsive part decays slower than 1/r2 the bound states approaching the threshold do not spread and eventually become bound states at the threshold. This means that such systems never reach supersizes, which would extend far beyond the effective range of attraction. The method presented here is applicable in the many-body case