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.