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The efimov effect of three-body schrödinger operators: Asymptotics for the number of negative eigenvalues

Published online by Cambridge University Press:  22 January 2016

Hideo Tamura
Affiliation:
Department of Mathematics, Ibaraki University Mito, Ibaraki, 310, Japan
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Extract

The Efimov effect is one of the most remarkable results in the spectral theory for three-body Schrödinger operators. Roughly speaking, the effect will be explained as follows: If all three two-body subsystems have no negative eigenvalues and if at least two of these two-body subsystems have resonance states at zero energy, then the three-body system under consideration has an infinite number of negative eigenvalues accumulating at zero. This remarkable spectral property was first discovered by Efimov [1] and the problem has been discussed in several physical journals. For related references, see, for example, the book [3]. The mathematically rigorous proof of the result has been given by the works [4, 8, 9]. The aim of the present work is to study the asymptotic distribution of these negative eigenvalues below zero (bottom of essential spectrum). Denote by N(E), E > 0, the number of negative eigenvalues less than – E. Then the main result obtained here is, somewhat loosely stating, that N(E) behaves like | log E | as E → 0. We first formulate precisely the main theorem and then make a brief comment on the recent related result obtained by Sobolev [7].

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1993

References

[1] Efimov, V., Energy levels arising from resonant two-body forces in a three-body systems, Phys. Lett., B 33 (1970), 563564.CrossRefGoogle Scholar
[2] Jensen, A. and Kato, T., Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583611.CrossRefGoogle Scholar
[3] Newton, R. G., Scattering Theory of Waves and Particles, 2nd ed., Springer Verlag, 1982.Google Scholar
[4] Ovchinnikov, Y. N. and Sigal, I. M., Number of bound states of three-body systems and Efimov’s effect, Ann. Phys., 123 (1979), 274295.CrossRefGoogle Scholar
[5] Reed, M. and Simon, B., Methods of Modern Mathematical Physics IV, Analysis of Operators, Academic Press, 1982.Google Scholar
[6] Simon, B., Large time behavior of the L norm of Schrödinger semigroups, J. Funct. Anal., 40(1981), 6683.CrossRefGoogle Scholar
[7] Sobolev, A. V., The Efimov effect: Discrete spectrum asymptotics, preprint, University of Toronto (1992).Google Scholar
[8] Tamura, H., The Efimov effect of three-body Schrödinger operators, J. Funct. Anal., 95(1991), 433459.CrossRefGoogle Scholar
[9] Yafaev, D. R., On the theory of the discrete spectrum of the three-particle Schrödinger operator, Math. USSR-Sb, 23 (1974), 535559.Google Scholar

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