Universal low-energy behavior in three-body systems

Abstract

We consider a pairwise interacting quantum 3-body system in 3-dimensional space with finite masses and the interaction term V 12 + λ(V 13 + V 23 ), where all pair potentials are assumed to be nonpositive. The pair interaction of the particles {1, 2} is tuned to make them have a zero energy resonance and no negative energy bound states. The coupling constant λ > 0 is allowed to take the values for which the particle pairs {1, 3} and {2, 3} have no bound states with negative energy.Let λ cr denote the critical value of the coupling constant such that E(λ) → −0 for λ → λ cr , where E(λ) is the ground state energy of the 3-body system. We prove the theorem, which states that near λ cr one has E(λ) = C(λ − λ cr )[ln(λ − λ cr )] −1 +h.t., where C is a constant and h.t. stands for "higher terms". This behavior of the ground state energy is universal (up to the value of the constant C), meaning that it is independent of the form of pair interactions.

Efimov Effect for a Three-Particle System with Two Identical Fermions

“…Other mathematical proofs of the effect were obtained by Ovchinnikov and Sigal in 1979 ([18]) and Tamura in 1991 ( [21]) using a variational approach based on the Born-Oppenheimer approximation (see also the related result in [14]). For more recent results on the subject see [12], [13]. It is worth mentioning that a mathematical proof of the geometric asymptotic law (1.1) is still lacking (see the conjecture discussed in [1]).…”