Communications in Mathematical Physics

, Volume 154, Issue 2, pp 347–376 | Cite as

Orthogonality and completeness of the Bethe Ansatz eigenstates of the nonlinear Schroedinger model

  • T. C. Dorlas
Article

Abstract

A rigorous proof is given of the orthogonality and the completeness of the Bethe Ansatz eigenstates of theN-body Hamiltonian of the nonlinear Schroedinger model on a finite interval. The completeness proof is based on ideas of C.N. Yang and C.P. Yang, but their continuity argument at infinite coupling is replaced by operator monotonicity at zero coupling. The orthogonality proof uses the algebraic Bethe Ansatz method or inverse scattering method applied to a lattice approximation introduced by Izergin and Korepin. The latter model is defined in terms of monodromy matrices without writing down an explicit Hamiltonian. It is shown that the eigenfunctions of the transfer matrices for this model converge to the Bethe Ansatz eigenstates of the nonlinear Schroedinger model.

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References

  1. 1.
    Babbitt, D., Thomas, L.E.: Commun. Math. Phys.54, 255–278 (1977)Google Scholar
  2. 2.
    Babbitt, D., Thomas, L.E.: J. Math. Phys.19, 1699–1704 (1978)Google Scholar
  3. 3.
    Babbit, D., Thomas, L.E.: J. Math. An. & Appl.72 305–328 (1979)Google Scholar
  4. 4.
    Babbit, D., Thomas, L.E., Gutkin: Lett Math. Phys.20, 91–99 (1990)Google Scholar
  5. 5.
    Baxter, R.J.: Ann. Phys.70, 323–337 (1972)Google Scholar
  6. 6.
    Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. New York: Academic Press 1982Google Scholar
  7. 7.
    Berg, M. van den, Lewis, J.T., Pulé, J.V.: Commun. Math. Phys.118, 61–85 (1988)Google Scholar
  8. 8.
    Berg, M. van den, Dorlas, T.C., Lewis, J.T., Pulé, J.V.: Commun. Math. Phys.127, 41–69 (1990)Google Scholar
  9. 9.
    Berg, M. van den, Dorlas, T.C., Lewis, J.T., Pulé, J.V.: Commun. Math. Phys.128, 231–245 (1990)Google Scholar
  10. 10.
    Bergknoff, H., Thacker, H.B.: Phys. Rev.D19, 3666–3681 (1979)Google Scholar
  11. 11.
    Bethe, H.: Zeits. f. Phys.71, 205–226 (1931)Google Scholar
  12. 12.
    Craemer, D.B., Thacker, H.B., Wilkinson, D.: Phys. Rev.D21, 1523–1528 (1980)Google Scholar
  13. 13.
    Destri, C., Lowenstein, J.H.: Nucl. Phys.B205, 369–385 (1982)Google Scholar
  14. 14.
    De Vega, H.J.: Int. J. Mod. Phys.A4, 2371–2463 (1989)Google Scholar
  15. 15.
    Dorlas, T.C., Lewis, J.T., Pulé, J.V.: Commun. Math. Phys.124, 365–402 (1989)Google Scholar
  16. 16.
    Faddeev, L.D.: Sov. Sc. Rev.C1, 107–155 (1980)Google Scholar
  17. 17.
    Faddeev, L.D., Sklyanin, E.K., Takhtadzhyan, L.A.: Theor. Math. Phys.40 (1979)Google Scholar
  18. 18.
    Faddeev, L.D., Takhtadzhyan, L.A.: Russ. Math. Surv.1 Google Scholar
  19. 19.
    Gaudin, M.: La fonction d'onde de Bethe, Paris: Masson, 1983Google Scholar
  20. 20.
    Girardeau, M.: J. Math. Phys.1, 516–523 (1960)Google Scholar
  21. 21.
    Izergin, A.G., Korepin, V.E.: Sov. Phys. Dokl.26, 653–654 (1981)Google Scholar
  22. 22.
    Izergin, A.G., Korepin, V.E.: Nucl. Phys.B205 [FS5], 401–413 (1982)Google Scholar
  23. 23.
    Izergin, A.G., Korepin, V.E., Smirnov, F.A.: Theor. Math. Phys.48, 773–776 (English) (1981)Google Scholar
  24. 24.
    Korepin, V.E.: Commun. Math. Phys.86, 391–418 (1982)Google Scholar
  25. 25.
    Landau, L.D., Lifschitz, E.M.: Statistical Physics, Vol. 5 of: ‘Course of Theoretical Physics’, London: Pergamon Press 1958Google Scholar
  26. 26.
    Lieb, E.H., Liniger, W.: Phys. Rev.130, 1605–1624 (1963)Google Scholar
  27. 27.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. (Revised and enlarged edition), New York: Academic Press 1980Google Scholar
  28. 28.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis and Self-Adjointness. New York: Academic Press 1975Google Scholar
  29. 29.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV: Analysis of Operators. New York: Academic Press 1978Google Scholar
  30. 30.
    Simon, B.: J. Funct. Anal.28, 377–385 (1978)Google Scholar
  31. 31.
    Sklyanin, E.K.: Sov. Phys. Dokl.24, 107–109 (1979)Google Scholar
  32. 32.
    Takahashi, M.: Progr. Theor. Phys.46, 401–415 (1971)Google Scholar
  33. 33.
    Thacker, H.B., Wilkinson, D.: Phys. Rev.D19, 3660–3665 (1979)Google Scholar
  34. 34.
    Thacker, H.B.: Rev. Mod. Phys.53, 253–285 (1981)Google Scholar
  35. 35.
    Thomas, L.E.: J. Math. Anal. & Appl.59, 392–414 (1977)Google Scholar
  36. 36.
    Yang, C.N., Yang, C.P.: J. Math. Phys.10, 1115–1122 (1969)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • T. C. Dorlas
    • 1
  1. 1.University College SwanseaSwanseaUK

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