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 $\lim_{n \to \infty}\int_{|\zeta| \leq R} |\psi_n (\zeta)|^2 d\zeta = 0$forall$R>0$. 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.