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Universal low-energy behavior in three-body systems

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 $E(λ) \to -0$ for $λ\to λ_{cr}$, 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.