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The Efimov effect of three-body Schrödinger operators: Asymptotics for the number of negative eigenvalues. (English) Zbl 0827.35099
The author considers a system of three particles which move in the three- dimensional space and interact with each other through a pair potential $$V_{jk} (r_j - r_k)$$ $$(1 \leq j < k \leq 3$$, $$r_j \in \mathbb{R}^3)$$. It is supposed that $$|V_{jk} (x) |\leq C(1 + |x |)^{-\rho}$$ for some $$\rho > 2$$ and every two-body subsystem Hamiltonian has no negative bound state energies and has a resonance state at zero energy. For three-body Schrödinger operator the formula $N(E) = C_0 |\log E |\bigl( 1 + o(1) \bigr), \quad E \to 0$ is proved where $$N(E)$$ is the number of eigenvalues less than $$- E$$.

##### MSC:
 35P25 Scattering theory for PDEs 47A10 Spectrum, resolvent 81U10 $$n$$-body potential quantum scattering theory
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##### References:
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