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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 \(- E\).

35P25 Scattering theory for PDEs
47A10 Spectrum, resolvent
81U10 \(n\)-body potential quantum scattering theory
Full Text: DOI
[1] DOI: 10.1070/SM1974v023n04ABEH001730 · Zbl 0342.35041 · doi:10.1070/SM1974v023n04ABEH001730
[2] DOI: 10.1016/0022-1236(91)90038-7 · Zbl 0761.35078 · doi:10.1016/0022-1236(91)90038-7
[3] The Efimov effect: Discrete spectrum asymptotics (1992) · Zbl 0785.35070
[4] DOI: 10.1016/0022-1236(81)90073-2 · Zbl 0478.47024 · doi:10.1016/0022-1236(81)90073-2
[5] Phys. Lett. B 33 pp 563– (1970)
[6] DOI: 10.1016/0003-4916(79)90339-7 · doi:10.1016/0003-4916(79)90339-7
[7] (1982)
[8] DOI: 10.1215/S0012-7094-79-04631-3 · Zbl 0448.35080 · doi:10.1215/S0012-7094-79-04631-3
[9] Analysis of Operators (1982)
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