Universal Angular Probability Distribution of Three Particles near Zero Energy Threshold

We study bound states of a 3--particle system in \(\mathbb{R}^3\) described by the Hamiltonian \(H(\lambda_n) = H_0 + v_{12} + \lambda_n (v_{13} + v_{23})\), where the particle pair \(\{1,2\}\) has a zero energy resonance and no bound states, while other particle pairs have neither bound states nor zero energy resonances. It is assumed that for a converging sequence of coupling constants \(\lambda_n \to \lambda_{cr}\) the Hamiltonian \(H(\lambda_n)\) has a sequence of levels with negative energies \(E_n\) and wave functions \(\psi_n\), where the sequence \(\psi_n\) totally spreads in the sense that \(\lim_{n \to \infty}\int_{|\zeta| \leq R} |\psi_n (\zeta)|^2 d\zeta = 0\) for all \(R>0\). We prove that for large \(n\) the angular probability distribution of three particles determined by \(\psi_n\) approaches the universal analytical expression, which does not depend on pair--interactions. The result has applications in Efimov physics and in the physics of halo nuclei.