Universal Angular Probability Distribution of Three Particles near Zero Energy Threshold

الأصل · EN

We study bound states of a 3--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 wave functions ψₙ, where the sequence ψₙ totally spreads in the sense that ₙ → ∞∫|ζ| ≤ ᵣ |ψₙ (ζ)|² dζ= 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.