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Classical and quantum-mechanical systems of Toda lattice type. I

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Abstract

The structure of the commutant of Laplace operators in the enveloping and “Poisson algebra” of certain generalized “ax +b” groups leads (in this article) to a determination of classical and quantum mechanical first integrals to generalized periodic and non-periodic Toda lattices. Certain new Hamiltonian systems of Toda lattice type are also shown to fit in this framework. Finite dimensional Lax forms for the (periodic) Toda lattices are given generalizing results of Flaschke.

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References

  1. Adler, M.: Invent. Math.50, 219–248 (1979)

    Google Scholar 

  2. Adler, M., van Moerbeke, P.: Adv. Math.38, 267–317 (1980)

    Google Scholar 

  3. Arnol'd, V.: Ann. Inst. Fourier (Grenoble)16, 319–361 (1966)

    Google Scholar 

  4. Arnol'd, V.: Mathematical methods of classical mechanics. Berlin, Heidelberg, New York: Springer 1978

    Google Scholar 

  5. Bogoyavlensky, O.I.: Commun. Math. Phys.51, 201–209 (1976)

    Google Scholar 

  6. Bourbaki, N.: Groupes et algèbres de Lie, Chaps. IV–VI (Éléments de mathématique, Fasc. XXXIV). Paris: Hermann 1968

    Google Scholar 

  7. Flaschka, H.: Phys. Rev. B9, 1924–1925 (1974)

    Google Scholar 

  8. Goodman, R., Wallach, N.R.: J. Funct. Anal.39, 199–279 (1980)

    Google Scholar 

  9. Helgason, S.: Differential geometry and symmetric spaces. New York: Academic Press 1962

    Google Scholar 

  10. Hénon, M.: Phys. Rev. B9, 1921–1923 (1974)

    Google Scholar 

  11. Kostant, B.: Invent. Math.48, 101–184 (1978)

    Google Scholar 

  12. Kostant, B.: Adv. Math.34, 195–338 (1979)

    Google Scholar 

  13. Moser, J.: Various aspects of integrable Hamiltonian systems. In: Dynamical systems. Progress in Mathematics, Vol. 8. Boston: Birkhäuser 1980

    Google Scholar 

  14. Miščenko, A.S., Fomenko, A.T.: Izv. Akad. Nauk. SSSR Ser. Mat.42, 396–415 (1978); Math. USSR Izv.12, 371–389 (1978)

    Google Scholar 

  15. Nelson, E., Stinespring, W.F.: Am. J. Math.81, 547–560 (1959)

    Google Scholar 

  16. Olshanetsky, M.A., Perelomov, A.M.: Invent. Math.54, 261–269 (1979)

    Google Scholar 

  17. Ratiu, T.: Involution theorems. In: Geometric methods in mathematical physics. Lecture Notes in Mathematics, Vol. 775. Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  18. Vergne, M.: Bull. Soc. Math. Fr.100, 301–335 (1972)

    Google Scholar 

  19. van Moerbeke, P.: Invent. Math.37, 45–81 (1976)

    Google Scholar 

  20. Wallach, N.R.: Symplectic geometry and fourier analysis. Lie Groups: History, Frontiers, and Applications. V. Brookline, Massachusetts: Math. Sci. Press 1977

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA

    Roe Goodman & Nolan R. Wallach

Authors
  1. Roe Goodman
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  2. Nolan R. Wallach
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Additional information

Communicated by A. Jaffe

Research partially supported by NSF grant MCS 79-03223

Research partially supported by NSF grant MCS 79-03153

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Goodman, R., Wallach, N.R. Classical and quantum-mechanical systems of Toda lattice type. I. Commun.Math. Phys. 83, 355–386 (1982). https://doi.org/10.1007/BF01213608

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