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 $V_{12}+\lambda (V_{13}+V_{23})$, 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 $\lambda >0$ is allowed to take the values for which the particle pairs $\{1,3\}$ and $\{2,3\}$ have no bound states with negative energy. Let ${\lambda }_{cr}$ denote the critical value of the coupling constant such that $E(\lambda )\to -0$ for $\lambda \to {\lambda }_{cr}$, where $E(\lambda )$ is the ground state energy of the 3-body system. We prove the theorem, which states that near ${\lambda }_{cr}$ one has $E(\lambda )=C(\lambda -{\lambda }_{cr})[\ln (\lambda -{\lambda }_{cr}){]}^{-1}+$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.

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Cited by in corpus (2)

• Efimov effect for a three-particle system with two identical fermions
• Rigorous derivation of the Efimov effect in a simple model