The Efimov effect of three-body Schrödinger operators: Asymptotics for the number of negative eigenvalues. (English) Zbl 0827.35099

Tamura, Hideo

Nagoya Math. J. 130, 55-83 (1993).

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 \(- E\).

Reviewer: S.L.Edelstein (Rostov-na-Donu)

MSC:

35P25	Scattering theory for PDEs
47A10	Spectrum, resolvent
81U10	\(n\)-body potential quantum scattering theory

Keywords:

Hamiltonian; bound state; resonance state; three-body Schrödinger operator

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