Universal Angular Probability Distribution of Three Particles near Zero Energy Threshold

Abstract

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 ${\lim }_{n\to \infty }{\int }_{|\zeta |\le R}|{\psi }_n(\zeta ){|}^{2}d\zeta =0$ 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.

References in corpus (3)

• On the Efimov Effect in Higher Partial Waves
• Bound States at Threshold resulting from Coulomb Repulsion
• Rigorous conditions for the existence of bound states at the threshold in the two-particle case

Cited by in corpus (2)

• Why there is no Efimov effect for four bosons and related results on the finiteness of the discrete spectrum
• Universal low-energy behavior in three-body systems