Universal angular probability distribution of three particles near zero-energy threshold

We study bound states of a three-particle system in R3 described by the Hamiltonian H(λn) = H0 + v12 + λn(v13 + v23), 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 λn → λcr the Hamiltonian H(λn) has a sequence of levels with negative energies En and wavefunctions ψn, where the sequence ψn totally spreads in the sense that limn→∞∫|ζ|⩽R|ψn(ζ)|2dζ = 0 for all R > 0. We prove that for large n the angular probability distribution of three particles determined by ψn 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. (paper)