Skip to main content
SpringerLink
Log in

Systems of Calogero-Moser Type

  • Chapter
Particles and Fields

Part of the book series: CRM Series in Mathematical Physics ((CRM))

Abstract

We survey results on Galilei-and Poincaré-invariant CalogeroMoser and Toda N-particle systems, both in the context of classical mechanics and of quantum mechanics. Special attention is given to integrability issues and interconnections between the various models. Action-angle and joint eigenfunction transforms are also considered, and some novel results on N = 2 eigenfunctions of hyperbolic Askey-Wilson type and of relativistic elliptic type are sketched.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. A. Olshanetsky and A. M. Perelomov. Classical integrable finite-dimensional systems related to Lie algebras. Phys. Rep., 71 (5): 313–400, 1981.

    Article  MathSciNet  ADS  Google Scholar 

  2. M. A. Olshanetsky and A. M. Perelomov. Quantum integrable systems related to Lie algebras. Phys. Rep.,94 (6): 313–404, 1983.

    Article  MathSciNet  ADS  Google Scholar 

  3. S. N. M. Ruijsenaars and H. Schneider. A new class of integrable systems and its relation to solitons. Ann. Phys., 170 (2): 370–405, 1986.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. S. N. M. Ruijsenaars. Complete integrability of relativistic CalogeroMoser systems and elliptic function identities. Commun. Math. Phys., 110 (2): 191–213, 1987.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. S. N. M. Ruijsenaars Finite-dimensional soliton systems. In B. Kupershmidt, ed., Integrable and Superintegrable Systems. World Scientific, Teaneck, NJ, pages 165–206, 1990.

    Chapter  Google Scholar 

  6. J. F. van Diejen. Commuting difference operators with polynomial eigenfunctions. Comp. Math., 95 (2): 183–233, 1995.

    MATH  Google Scholar 

  7. J. F. van Diejen. Integrability of difference Calogero-Moser systems. J. Math. Phys., 35 (6): 2983–3004, 1994.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. J. F. van Diejen. Difference Calogero-Moser systems and finite Toda chains. J. Math. Phys., 36 (3): 1299–1323, 1995.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. V. I. Arnol’d. Mathematical Methods of Classical Mechanics,volume 60 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1978.

    Google Scholar 

  10. W. Thirring. Lehrbuch der mathematischen Physik. I. Klassische dynamische Systeme. Springer-Verlag, Vienna, 1977.

    Google Scholar 

  11. R. Abraham and J. E. Marsden. Foundations.of Mechanics, 2nd edition. Benjamin-Cummings, Reading, MA, 1978.

    Google Scholar 

  12. E. T. Whittaker and G. N. Watson. A Course of Modern Analysis. Cambridge Univ. Press, Cambridge, 1973.

    Google Scholar 

  13. M. Reed and B. Simon. Methods.of Modern Mathematical Physics III. Scattering Theory. Academic, New York, 1979.

    Google Scholar 

  14. B. Sutherland. Exact results for a quantum many-body problem in one dimension II. Phys. Rev., A5: 1372–1376, 1972.

    ADS  Google Scholar 

  15. G. J. Heckman. Root systems and hypergeometric functions II. Comp. Math.,64 (3): 353–373, 1987.

    MathSciNet  MATH  Google Scholar 

  16. I. G. Macdonald. Orthogonal polynomials associated with root systems. preprint, 1988.

    Google Scholar 

  17. I. G. Macdonald. Orthogonal polynomials associated with root systems. In P. Nevai, ed., Orthogonal Polynomials, (Columbus, OH, 1989), volume C294 of NATO ASI, 1990. Kluwer, Dordrecht, pages 311–318.

    Chapter  Google Scholar 

  18. M. F. E. de Jeu. The Dunkl transform. Invent. Math., 113 (1): 147–162, 1993.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. P. I. Etingof and A. A. Kirillov, Jr. Macdonald’s polynomials and representations of quantum groups. Math. Res. Lett., 1 (3): 279–296, 1994.

    MathSciNet  MATH  Google Scholar 

  20. P. I. Etingof and A. A. Kirillov, Jr. Representations of affine Lie algebras, parabolic differential equations, and Lamé functions. Duke Math. J., 74 (3): 585–614, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Noumi. Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces. Adv. Math., to appear.

    Google Scholar 

  22. I. Cherednik. Difference-elliptic operators and root systems. Internat. Math. Res. Notices, 1995 (1): 43–58, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Askey and J. Wilson. Some Basic Hypergeometric Orthogonal Polynomials that Generalize Jacobi Polynomials,volume 319 of Mem. Amer. Math. Soc. Amer. Math. Soc., Providence, RI, 1985.

    Google Scholar 

  24. R. Floreanini and L. Vinet. Quantum algebras and q-special functions. Ann. Phys., 221 (1): 53–70, 1993.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. T. H. Koornwinder. Orthogonal polynomials in connection with quantum groups. In P. Nevai, ed., Orthogonal Polynomials, (Columbus, OH, 1989), volume C294 of NATO ASI, 1990. Kluwer, Dordrecht, pages 257–292.

    Chapter  Google Scholar 

  26. T. H. Koornwinder. Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group. SIAM J. Math. Anal., 24 (3): 795–813, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  27. N. J. Vilenkin and A. U. Klimyk. Representation of Lie Groups and Special Functions III. Classical and Quantum Groups and Special Functions, volume 75 of Mathematics and Its Applications (Soviet Series). Kluwer, Dordrecht, 1992.

    Google Scholar 

  28. S. N. M. Ruijsenaars. Generalized Lamé functions. J. Math. Phys., to appear.

    Google Scholar 

  29. S. N. M. Ruijsenaars. Relativistic Calogero-Moser systems and soli-tons. In M. Ablowitz, B. Fuchssteiner, and M. Kruskal, eds., Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, (Oberwolfach, 1986), 1987. World Scientific, Singapore, pages 182–190.

    Google Scholar 

  30. S. N. M. Ruijsenaars. Action-angle maps and scattering theory for some finite-dimensional integrable systems II. Solitons, antisolitons, and their bound states. Publ. RIMS, Kyoto Univ., 30 (6): 865–1008, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  31. S. N. M. Ruijsenaars. First-order analytic difference equations and integrable quantum systems. J. Math. Phys., 38: 1069–1146, 1997.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. G. Felder and A. P. Veselov. Shift operators for the quantum Calogero-Sutherland problems via the Knizhnik-Zamolodchikov equation. Commun. Math. Phys., 160 (2): 259–273, 1994.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. J. Moser. Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math., 16: 197–220, 1975.

    Article  ADS  MATH  Google Scholar 

  34. I. M. Krichever. Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles. Func. Anal. Appl., 14 (4): 282–290, 1980.

    Article  MathSciNet  Google Scholar 

  35. V. I. Inozemtsev. The finite Toda lattices. Commun. Math. Phys., 121 (4): 629–638, 1989.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. B. Sutherland. An introduction to the Bethe ansatz. In B. S. Shastry, S. S. Jha, and V. Singh, eds., Exactly Solvable Problems in Condensed Matter and Relativistic Field Theory, (Panchgani, 1985), volume 242 of Lecture Notes in Physics, 1985. Springer-Verlag, Berlin, pages 1–95.

    Chapter  Google Scholar 

  37. S. N. M. Ruijsenaars. Relativistic Toda systems. Commun. Math. Phys., 133 (2): 217–247, 1990.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. G. Frobenius. Über die elliptischen functionen zweiter art. J. Reine und.Angew. Math., 93: 53–68, 1882.

    MATH  Google Scholar 

  39. A. K. Raina. An algebraic geometry study of the b-c system with arbitrary twist fields and arbitrary statistics. Commun. Math. Phys., 140 (2): 373–397, 1991.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. S. N. M. Ruijsenaars. Action-angle maps and scattering theory for some finite-dimensional integrable systems III. Sutherland type systems and their duals. Publ. RIMS, Kyoto Univ., 31 (2): 247–353, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  41. J. von Neumann Mathematical Foundations of Quantum Mechanics. Princeton Univ. Press, Princeton, NJ, 1955.

    MATH  Google Scholar 

  42. M. Reed and B. Simon. Methods of Modern Mathematical Physics I. Functional Analysis. Academic, New York, 1972.

    Google Scholar 

  43. M. Reed and B. Simon. Methods of Modern Mathematical Physics II. Fourier Analysis,Self-Adjointness. Academic, New York, 1975.

    MATH  Google Scholar 

  44. M. Reed and B. Simon. Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic, New York, 1978.

    MATH  Google Scholar 

  45. W. Thirring. Lehrbuch der mathematischen Physik. III. Quantenmechanik von.Atomen und Molekülen. Springer-Verlag, Vienna, 1979.

    Google Scholar 

  46. E. M. Opdam. Root systems and hypergeometric functions III. Comp. Math., 67 (1): 21–49, 1988.

    MathSciNet  MATH  Google Scholar 

  47. E. M. Opdam. Root systems and hypergeometric functions IV. Comp. Math.,67 (2): 191–209, 1988.

    MathSciNet  MATH  Google Scholar 

  48. G. J. Heckman and E. M. Opdam. Root systems and hypergeometric functions I. Comp. Math.,64 (3): 329–352, 1987.

    MathSciNet  MATH  Google Scholar 

  49. T. Oshima and H. Sekiguchi. Commuting families of differential operators invariant under the action of a Weyl group. J. Math. Sci. Univ. Tokyo, 2 (1): 1–75, 1995.

    MathSciNet  MATH  Google Scholar 

  50. K. Sawada and T. Kotera. Integrability and a solution for the one-dimensional N-particle system with inversely quadratic pair potentials. J. Phys. Soc. Japan, 39 (6): 1614–1618, 1975.

    Article  MathSciNet  ADS  Google Scholar 

  51. S. Wojciechowski. Involutive set of integrals for completely integrable many-body problems with pair interaction. Lett. Nuovo Cim., 18: 103–107, 1977.

    Article  Google Scholar 

  52. A. T. Fomenko. Integrability and Nonintegrability in Geometry and Mechanics, volume 31 of Mathematics and Its Applications (Soviet Series). Kluwer, Dordrecht, 1988.

    Google Scholar 

  53. S. N. M. Ruijsenaars. Action-angle maps and scattering theory for some finite-dimensional integrable systems I. The pure soliton case. Commun. Math. Phys., 115 (1): 127–165, 1988.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  54. R. Goodman and N. R. Wallach. Classical and quantum-mechanical systems of Toda lattice-type III. Joint eigenfunctions of the quantized systems. Commun. Math. Phys., 105 (3): 473–509, 1986.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. J. Dittrich and V. I. Inozemtsev. On the structure of eigenvectors of the multidimensional Lamé operator. J. Phys A: Math. Gen., 26 (16): L753–L756, 1993.

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Authors
  1. S. N. M. Ruijsenaars
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, V6T 1Z1, Canada

    Gordon Semenoff

  2. Centre de Recherches Mathématiques, Université de Montréal, Montreal, Québec, H3C 3J7, Canada

    Luc Vinet

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Science+Business Media New York

About this chapter

Cite this chapter

Ruijsenaars, S.N.M. (1999). Systems of Calogero-Moser Type. In: Semenoff, G., Vinet, L. (eds) Particles and Fields. CRM Series in Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1410-6_7

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-1410-6_7

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7133-8

  • Online ISBN: 978-1-4612-1410-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation