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On the existence of the \(N\)-body Efimov effect. (English) Zbl 1059.81061
The author considers the Efimov effect for the \(N\)-body problem with \(N\geq 4\) for the Schrödinger equation in \({\mathbb R}{^ 3},\) where the potential consists of the sum of two-body interaction terms depending only on the separation distance. Under appropriate restrictions, it is shown that there may be an infinite number of discrete eigenvalues below \(E_ 0\) for the whole system as a result of the contributions from the eigenvalues and resonances at \(E_ 0\) of the \((N-1)\)-particle system, where \(E_ 0\) is the threshold energy separating the discrete and continuous spectra.
The proof uses the ideas from A. V. Sobolev [Commun. Math. Phys. 156, No. 1, 101–126 (1993; Zbl 0785.35070)] and exploits the conditions on the finiteness of number of discrete eigenvalues by W. D. Evans and R. T. Lewis [Trans. Am. Math. Soc. 322, No. 2, 593–626 (1990; Zbl 0732.35062)].

MSC:
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81U10 \(n\)-body potential quantum scattering theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables
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