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Solvable Many-Body Problems

  • F. Calogero
Conference paper
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 47)

Abstract

The material presented in this lecture has been reported in print several times and I don’t think it would make any sense for me to write it up once more. The only message I wish to record in the proceedings of this workshop is that there exist a number of many-body problems, mainly one-dimensional ones, that can be solved in rather explicit detail. Some of them can be treated both in the classical and quantal context; others have been investigated so far only in a classical context. As an example of the first kind let me mention the n-body problem of n1+n2=n nonrelativistic particles of two types moving on the line and interacting pairwise via the potentials
$$ {V_e}(x) = {g^2}{a^2}/{\sinh^2}\left( {ax} \right) $$
(1a)
$$ {V_d}(x) = - {g^2}{a^2}/{\cosh^2}\left( {ax} \right) $$
(1b)
with Ve acting between particles of the same type, Vd between particles of different types. Note that Ve is repulsive and singular at zero separation, while Vd is attractive and nonsingular and may therefore yield “molecules” made up of two particles of different types bound together. As an example of the second kind let me mention the fairly general equations of motion Open image in new window

References

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    F. Calogero: Integrable Dynamical Systems and Related Mathematical Results. In: Proceedings of the School and Workshop of Nonlinear Phenomena (K.B. Wolf, ed.), Oaxtepec, Mexico, November 29 — December 17, 1982; to appear in the series Lecture Notes in Physics, Springer, 1983.Google Scholar
  2. 2.
    F. Calogero: Integrable Many-Body Problems and Related Mathematical Results. In: Fundamental Problems in Statistical Mechanics V, Proceedings of the 1980 Enschede Summer School, North Holland, Amsterdam, 1981, pp.151–164.Google Scholar
  3. 3.
    F. Calogero: Solvable Many-Body Problems and Related Mathematical Findings (and Conjectures). In: Bifurcation Phenomena in Mathematical Physics and Related Topics (C. Bardos and D. Bessis, eds.), Reidel, Dordrecht, 1980.Google Scholar
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    M.A. Olshanetsky and A.M. Perelomov: Classical Integrable Finite-Dimensional Systems Related to Lie Algebras. Physics Reports 71, 313–400 (1981).MathSciNetADSCrossRefGoogle Scholar
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    M.A. Olshanetsky and A.M. Perelomov: Quantal Integrable Systems Related to Lie Algebras. Physics Reports (in press).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • F. Calogero
    • 1
  1. 1.Dipartimento di Fisica, Istituto Nazionale di Fisica Nucleare, Sezione di RomaUniversità di Roma “La Sapienza”RomaItaly

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