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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 $H(λ_n) = H_0 + v_{12} + λ_n (v_{13} + v_{23})$, 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 $λ_n \to λ_{cr}$ the Hamiltonian $H(λ_n)$ has a sequence of levels with negative energies $E_n$ and wave functions $ψ_n$, where the sequence $ψ_n$ totally spreads in the sense that $\lim_{n \to \infty}\int_{|ζ| \leq R} |ψ_n (ζ)|^2 dζ= 0$ for all $R>0$. We prove that for large $n$ the angular probability distribution of three particles determined by $ψ_n$ 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.