Progress in Particle Physics pp 679-709 | Cite as

# The N-Body Problem

Conference paper

## Abstract

Two years ago, in a lecture given at the Schladming Conference 1972, I reviewed the present status of the quantum mechanical three-body problem [1]. It was the main intention of this survey to illustrate the way in which the properties of the integral equations, studied in this field, are related to basic concepts of multichannel collision theory. Such a consideration makes the typical difficulties of older attempts rather transparent. Moreover, the modern approaches initiated by Faddeev’s work [2], are easily understood as natural consequences of general features of the multichannel theory.

## Keywords

Body Case Faddeev Equation Heisenberg Picture Pole Approximation Single Channel Case
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.W. Sandhas, The Three-Body Problems, in Elementary Particle Physics, ed.: P. Urban (Acta Physica Austriaca, Suppl. IX, 57 (1972)).Google Scholar
- 2.L. D. Faddeev, Mathematical Aspects of the Three-Body Problem in the Quantum Scattering Theory (English translation: Israel Program for Scientific Translation, Jerusalem, 1965 ).Google Scholar
- 3.Compare also: E. 0. Alt, P. Grassberger, W. Sandhas, JIRN, E4–6688, Dubna, 1972. See furthermore p. 299 of Few Particle Problems, ed.: I. Slaus et al., North-Holland, 1972 (Proceedings of the Los Angeles Conference 1972 ).Google Scholar
- 4.The definition of
^{2(+)}by a time limit is given in Eq. (2.9) of Ref. 1.Google Scholar - 5.Without going into any details we recall that in the Heisenberg picture the relevant operators have to be sandwiched between eigenstates of the total Hamiltonian. These, however, are in our case the scattering statesGoogle Scholar
- 6.E. O. Alt, P. Grassberger, W. Sandhas, Nucl. Phys. B2, 167 (1967).CrossRefADSGoogle Scholar
- 7.L. D. Faddeev, Soviet Phys. - JETP 12, 1014 (1961).MathSciNetGoogle Scholar
- 8.A detailed investigation, sketched in Ref. 1, shows that the uniqueness of the Faddeev equations is a consequence of the fact that their algebraic structure corresponds exactly to decisive aspects of the multichannel collision theory. The original, more technical proof of uniqueness [2] was based on methods of integral equations theory. From both points of view it is important that the expression (3.8) represents an exchange potential, different from zero only for a 6, or, in other words, that the kernel of the system of equations (3.4) has no diagonal elements.Google Scholar
- 9.P. Grassberger, W. Sandhas, Nucl. Phys. B2, 181 (1967).CrossRefADSGoogle Scholar
- 10.E. O. Alt, P. Grassberger, W. Sandhas, Phys. Rev. Cl, 85 (1970).Google Scholar
- 11.O. A. Yakubovsky, Sov. J. Nucl. Phys. 5, 937 (1967).Google Scholar
- L. D. Faddeev, Three-Body Problem in Nuclear and Particle Physics, ed.: J. S. C. McKee a.P.M. Rolph (North-Holland, Amsterdam, 1970 ). Compare alsoGoogle Scholar
- K. Hepp, Helv. Phys. Acta 42, 425 (1969).MathSciNetGoogle Scholar
- 12.An incomplete list of further publications is: S. Weinberg, Phys. Rev., 133, B232 (1964);Google Scholar
- L. Rosenberg, Phys. Rev., 140, B217 (1965);CrossRefADSMathSciNetGoogle Scholar
- A. N. Mitra, J. Gillespie, R. Sugar, N. Panchapakesan, Phys. Rev., 140, B1336 (1965);CrossRefADSMathSciNetGoogle Scholar
- V. V. Komarov, A. N. Popova, Nucl. Phys. 69, 253 (1965); Phys. Lett., 28B, 476 (1969);Google Scholar
- N. Mishima, Y. Takahashi, Progr. Theor. Phys. 35, 440 (1966);CrossRefADSGoogle Scholar
- I. Weyers, Phys. Rev., 145, 1236 (1966); Phys. Rev. 151, 1159 (1966);Google Scholar
- R. Omnes, Phys. Rev. 165, 1265 (1968)CrossRefADSGoogle Scholar
- I. Sloan, Phys. Rev. C6, 1945 (1972);ADSMathSciNetGoogle Scholar
- G. Bencze, Nucl. Phys. A210, 568 (1973);CrossRefGoogle Scholar
- A. N. Mitra, Flinders-preprint: FUPH-R-87Google Scholar
- É.F. Redish, Saclay-preprint: DPh. T. 74/3.Google Scholar
- 13.V. F. Kharchenko, Few Particle Problems, ed. I. Slaus et al., North-Holland, 1972 (Proceedings of the Los Angeles Conference 1972 ).Google Scholar

## Copyright information

© Springer-Verlag 1974