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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)].

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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[1] Ahia, F., Finiteness of the discrete spectrum of \((N⩾ 3)N\)-body Schrödinger operators, which have some determinate subsystems that are virtual at the bottom of the continuum, J. math. phys., 33, 1, 189-202, (1992)
[2] Albeverio, S.; Høegh-Krohn, R.; Wu, T.T., A class of exactly solvable three-body quantum mechanical problems and the universal low energy behavior, Phys. lett. A, 83, 3, 105-109, (1981)
[3] Albeverio, S.; Lakaev, S.; Makarov, K.A., The Efimov effect and an extended szegö-kac limit theorem, Lett. math. phys., 43, 1, 73-85, (1998) · Zbl 0903.45003
[4] Albeverio, S.; Makarov, K.A., Limit behaviour in a singular perturbation problem, regularized convolution operators and the three-body quantum problem. differential and integral operators, Oper. theory adv. appl., 102, 1-10, (1998), (Regensburg, 1995) · Zbl 0903.47039
[5] Amado, R.D.; Greenwood, F.C., There is no Efimov effect for four or more particles, Phys. rev. D, 7, 3, 2517-2519, (1973)
[6] R.D. Amado, J.V. Noble, On Efimov’s effect: a new pathology of three-particle systems, Phys. Lett. B 35 (1971) 25-27; II. Phys. Lett. D 5(3) (1972) 1992-2002.
[7] Birman, M.S.; Solomyak, M.Z., Spectral asymptotics of non smooth elliptic operators II, Trans. Moscow math. soc., 28, 1-32, (1973) · Zbl 0296.35066
[8] J.F. Brasche, Generalized Schrödinger operators, an inverse problem in spectral analysis and the Efimov effect, in: Stochastic Processes, Physics and Geometry (Ascona and Locarno, 1988), World Scientific Publishing, Teaneck, NJ, 1990, pp. 207-244.
[9] Coutinho, F.A.B.; Perez, J.F.; Wreszinski, W.F., A variational proof of the Thomas effect, J. math. phys., 36, 4, 1625-1635, (1995) · Zbl 0829.47061
[10] Cycon, H.L.; Froese, R.G.; Kirsch, W.; Simon, B., Schrödinger operators, with application to quantum mechanics and global geometry, (1984), Springer Berlin
[11] Efimov, V., Energy levels arising from resonant two-body forces in a three-body systems, Phys. lett. B, 33, 563-564, (1970)
[12] Evans, W.D.; Lewis, R.T., N-body Schrödinger operators with finitely many bound states, Trans. amer. math. soc., 322, 2, 593-626, (1990) · Zbl 0732.35062
[13] Evans, W.D.; Lewis, R.T.; Saito, Y., The agmon spectral function for molecular Hamiltonians with symmetry restrictions, Proc. roy. soc. London A, 440, 621-638, (1993) · Zbl 0804.35028
[14] U. Grenander, G. Szegö, Toeplitz Forms and their Applications, University of California Press, Berkeley, Los Angeles, 1958.
[15] Jensen, A.; Kato, T., Spectral properties of Schrödinger operators and time decay of wave functions, Duke math. J., 46, 583-611, (1979) · Zbl 0448.35080
[16] Karner, G., Many-body systems and the Efimov effect, Few-body systems, 3, 1, 7-25, (1987)
[17] Klaus, M.; Simon, B., Coupling constant thresholds in nonrelativistic quantum mechanics II. two cluster thresholds in N-body systems, Commun. math. phys., 78, 153-168, (1980/1981) · Zbl 0462.35082
[18] S.N. Lakaev, On the Efimov effect in a system of three identical quantum particles (Russian), Funk. Anal. Prilozhen. 27(3) (1993) 15-28, 95; translation in Funct. Anal. Appl. 27(3) (1993) 166-175.
[19] K.A. Makarov, V.V. Melezhik, Two sides of a coin: the Efimov effect and collapse in a three-body system with point interactions. I. (Russian), Teoret. Mat. Fiz. 107(3) (1996) 415-432; translation in Theoret. Math. Phys. 107(3) (1996) 755-769 (1997). · Zbl 1041.81526
[20] Newton, R.G., Noncentral potentialsthe generalized Levinson theorem and the structure of the spectrum, J. math. phys., 18, 1582-1588, (1977)
[21] Newton, R.G., Scattering theory for waves and particles, (1982), Springer Berlin
[22] Ovchinnikov, Yu.N.; Sigal, I.M., Number of bound states of three-body systems and Efimov’s effect, Ann. phys., 123, 2, 274-295, (1979)
[23] Reed, M.; Simon, B., Methods of modern mathematical physics, III. scattering theory, (1978), Academic Press New York
[24] Simon, B., Large time behavior of the Lp norm of Schrödinger semigroups, J. funct. anal., 40, 66-83, (1981) · Zbl 0478.47024
[25] Sobolev, A.V., The Efimov effect. discrete spectrum asymptotics, Commun. math. phys., 156, 101-126, (1993) · Zbl 0785.35070
[26] Tamura, H., The Efimov effect of three-body Schrödinger operators, J. funct. anal., 95, 2, 433-459, (1991) · Zbl 0761.35078
[27] Tamura, H., The Efimov effect of three-body Schrödinger operatorsasymptotics for the number of negative eigenvalues, Nagoya math. J., 130, 55-83, (1993) · Zbl 0827.35099
[28] Vugal’ter, S., Absence of the Efimov effect in a homogeneous magnetic field, Lett. math. phys., 37, 1, 79-94, (1996) · Zbl 0848.47045
[29] Vugal’ter, S.A.; Zhislin, G.M., On the discrete spectrum of Schrödinger operators of multiparticle systems with two-particle virtual levels, Dolk. akad. nauk. SSSR, 267, 4, 784-786, (1982) · Zbl 0561.35022
[30] Vugal’ter, S.A.; Zhislin, G.M., On the finiteness of discrete spectrum in the n-particle problem, Rep. math. phys., 19, 1, 39-90, (1984) · Zbl 0581.46063
[31] X.P. Wang, Asymptotic behavior of the resolvent for N-body Schrödinger operators near a threshold, Ann. H. Poincaré, in press.
[32] Yafaev, D.R., On the theory of the discrete spectrum of the three-particle Schrödinger operator, Math. USSR-sb., 23, 535-559, (1974) · Zbl 0342.35041
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