zbMATH — the first resource for mathematics

The Efimov effect. Discrete spectrum asymptotics. (English) Zbl 0785.35070
A three-particle Schrödinger operator \(H\) with short-range pair potentials is considered. It is supposed that two-particle subsystems do not have negative eigenvalues and at least two of them have zero-energy resonances. In this case the discrete spectrum of \(H\) is infinite (this is called the Efimov’s effect).
It is shown that the number \(N(z)\) of bound states of \(H\) below \(z<0\) has the asymptotics \(N(z)\sim a|\ln| z| |\) where the coefficient \(a\) depends only on masses of particles.

35P20 Asymptotic distributions of eigenvalues in context of PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
Full Text: DOI
[1] Albeverio, S., Høegh-Krohn, R., Wu, T. T.: A class of exactly solvable three-body quantum mechanical problems and the universal low energy behaviour. Phys. Lett. A.83, 3, 105–109 (1981) · doi:10.1016/0375-9601(81)90507-7
[2] Amado, R.D., Noble, J.V.: On Efimov’s Effect: A new pathology of three-particle systems. Phys. Lett. B35, 25–27 (1971); II, Phy. Lett D (3) 5, 1992–2002 (1972) · doi:10.1016/0370-2693(71)90429-1
[3] Birman, M.Sh., Solomyak, M.Z.: Spectral theory of selfadjoint operators in Hilbert space. Dordrecht: D. Reidel P.C., 1987 · Zbl 0744.47017
[4] Efimov, V.: Energy levels arising from resonant two-body forces in a three-body system. Phys. Lett. B33, 563–564 (1970) · doi:10.1016/0370-2693(70)90349-7
[5] Erdélyi, A., (Ed.): Higher transcendental functions. Vol. 2, New York, Toronto, London: McGraw-Hill 1953 · Zbl 0051.30303
[6] Faddeev, L.D.: Mathematical aspects of the three-body problem in the quantum scattering theory. Trudy Mat. Inst. Steklov.69 (1963) (Russian)
[7] Faddeev, L.D., Merkuriev, S.P: Scattering theory. Leningrad: Nauka, 1989 (Russian) · Zbl 0797.47005
[8] Grenander, U., Szego, G.: Toeplitz forms and their applications. 2nd ed., New York: Chelsea P.C., 1984 · Zbl 0611.47018
[9] Jensen, A., Kato, T.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke. Math. J.46, 5, 583–611 (1979) · Zbl 0448.35080 · doi:10.1215/S0012-7094-79-04631-3
[10] Ovchinnikov, Yu. N., Sigal, I.M.: Number of bound states of three body systems and Efimov’s effect. Ann. Phys.123, 274–295 (1979) · doi:10.1016/0003-4916(79)90339-7
[11] Tamura, H.: The Efimov effect of three-body Schrödinger operators. J. Funct. Anal.95, 433–459 (1991) · Zbl 0761.35078 · doi:10.1016/0022-1236(91)90038-7
[12] 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 · doi:10.1070/SM1974v023n04ABEH001730
[13] Yafaev, D.R.: On the zero resonance for the Schrödinger equation. Notes of LOMI Seminars51 (1975) (Russian)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.