Skip to main content
SpringerLink
Log in

Quantum version of the method of inverse scattering problem

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

This paper is a developed and consecutive account of a quantum version of the method of the inverse scattering problem on the example of the nonlinear Schrödinger equation. The method of R-matrices developed by the author is given basic consideration. The generating functions of quantum integrals of motion and action-angle variables for the quantum nonlinear Schrödinger equation are obtained. There is also described a classical version of the method of R-matrices.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature cited

  1. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: Method of the Inverse Problem [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  2. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-de Vries equation,” Phys. Rev. Lett.,19, 1095–1097 (1967).

    Google Scholar 

  3. P. D. Lax, “Integrals of nonlinear evolutional equations and isolated waves,” Matematika (collection of translations of foreign papers),13, No. 5, 128–150 (1967).

    Google Scholar 

  4. V. E. Zakharov and L. D. Faddeev, “Korteweg-de Vries equation —completely integrable Hamiltonian system,” Funkts. Anal. Prilozhen.,5, No. 4, 18–27 (1971).

    Google Scholar 

  5. H. Bethe, “Zur Theorie der Metalle. I. Eigenwerte und Eigenfunctionen der linearen Atomkette,” Z. Phys.,71, 205–226 (1931).

    Google Scholar 

  6. L. Onsager, “Crystal statistics. I. A two-dimensional model with an order-disorder transition,” Phys. Rev.,65 (1944).

  7. R. J. Baxter, “Partition function of the eight-vertex lattice model,” Ann. Phys.,70, No. 1, 193–288 (1972).

    Google Scholar 

  8. R. J. Baxter, “One-dimensional anisotropic Heisenberg chain,” Ann. Phys.,70, No. 2, 323–337 (1972).

    Google Scholar 

  9. V. E. Zahkarov, L. A. Takhtadzhyan, and L. D. Faddeev, “Complete description of solutions of the ‘sin-Gordon’ equation,” Dokl. Akad. Nauk SSSR,219, No. 6, 1334–1337 (1974).

    Google Scholar 

  10. V. E. Zakharov and A. V. Mikhailov, “Relativistically invariant two-dimensional models of the theory of fields, integrable by the method of the inverse problem,” Zh. Eksp. Teor. Fiz.,74, No. 6, 1953–1973 (1978).

    Google Scholar 

  11. P. P. Kulish, S. V. Manakov, and L. D. Faddeev, “Comparison of exact quantum and quasiclassical answers for the nonlinear Schrödinger equation,” Teor. Mat. Fiz.,28, No. 1, 38–45 (1976).

    Google Scholar 

  12. L. D. Faddeev and V. E. Korepin, “Quantum theory of solitons,” Phys. Reports,42C, No. 1, 1–87 (1978).

    Google Scholar 

  13. E. K. Sklyanin and L. D. Faddeev, “Quantum mechanical approach to completely integrable models of field theory,” Dokl. Akad. Nauk SSSR,243, No. 6, 1430–1433 (1978).

    Google Scholar 

  14. E. K. Sklyanin, “Method of the inverse scattering problem and quantum nonlinear Schrödinger equation,” Dokl. Akad. Nauk SSSR,244, No. 6, 1337–1341 (1978).

    Google Scholar 

  15. E. K. Sklyanin, “Quantum version of the method of the inverse scattering problem,” Candidates Dissertation, Leningrad Branch of the Mathematics Institute (1980).

  16. L. D. Faddeev, “Quantum completely integrable models of field theory,” in: Problems of Quantum Field Theory (Memoirs of the International Conference on Nonlocal Field Theories, Alushta, 1979) [in Russian], Dubna (1979), pp. 249–299.

  17. E. K. Sklyanin, L. A. Takhtadzhyan, and L. D. Faddeev, “Quantum method of the inverse problem I,” Teor. Mat. Fiz.,40, No. 2, 194–220 (1979).

    Google Scholar 

  18. P. P. Kulish and E. K. Sklyanin, “Quantum inverse scattering method and the Heisenberg ferromagnet,” Phys. Lett. A,70, Nos. 5–6, 461–463 (1979).

    Google Scholar 

  19. L. A. Takhtadzhyan and L. D. Faddeev, “Quantum method of the inverse problem and the XYZ-model of Heisenberg,” Usp. Mat. Nauk,34, No. 5, 13–63 (1979).

    Google Scholar 

  20. P. P. Kulish, “Generalized Bethe Ansatz and the quantum method of the inverse problem,” Preprint Leningrad Branch of the Mathematics Institute P-3-79, Leningrad (1979).

  21. P. P. Kulish and N. Yu. Reshetikhin, “Generalized Heisenberg ferromagnet and the Groos-Neveu model,” Preprint Leningrad Branch of the Mathematics Institute E-4-79, Leningrad (1979).

  22. H. B. Thacker and D. Wilkinson, “The inverse scattering transform as an operator method in quantum field theory,” Phys. Rev.,D19, No. 12, 3660–3665 (1979).

    Google Scholar 

  23. D. B. Creamer, H. B. Thacker, and D. Wilkinson, “Gelfand-Levitan method for operator fields,” Preprint Fermilab-Pub-79/75-THY, September, 1979.

  24. D. B. Creamer, H. B. Thacker, and D. Wilkinson, “Quantum Gelfand-Levitan method as a generalized Jordan-Wigner transformation,” Fermilab-Pub-80/17-THY, January, 1980.

  25. D. B. Creamer, H. B. Thacker, and D. Wilkinson, “Statistical mechanics of an exactly integrable system,” Fermilab-Pub-80/25-THY, February, 1980.

  26. J. Hönerkamp, P. Weber, and A. Wiesler, “On the connection between the inverse transform method and the exact quantum eigenstates,” Nucl. Phys.,B152, No. 2, 266 (1979).

    Google Scholar 

  27. A. Wiesler, “Rigorous proof of “Bethe's hypothesis” from inverse scattering transformation,” Preprint. Univ. Freiburg THEP 79/8, October, 1979.

  28. H. Grosse, “On the construction of Möller operators for the nonlinear Schrödinger equation,” Phys. Lett.,86B, Nos. 3–4, 267–271 (1979).

    Google Scholar 

  29. E. K. Sklyanin, “On complete integrability of the Landau-Lifshits equation,” Preprint, Leningrad Branch of the Mathematics Institute, E-3-79, Leningrad (1979).

    Google Scholar 

  30. V. E. Zakharov and S. V. Manakov, “Complete integrability of the nonlinear Schrödinger equation,” Teor. Mat. Fiz.,19, No. 3, 332–343 (1974).

    Google Scholar 

  31. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional automodulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz.,61, No. 1, 118–134 (1971).

    Google Scholar 

  32. L. A. Takhtadzhyan, “Hamiltonian systems connected with Dirac's equation,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,37, 66–76 (1973).

    Google Scholar 

  33. F. A. Berezin, G. P. Pokhil, and V. M. Finkel'berg, “Schrödinger's equation for systems of one-dimensional particles with pointwise interactions,” Vestn. Mosk. Gos. Univ., Ser. Mat., Mekh., No. 1, 21–28 (1964).

    Google Scholar 

  34. J. B. McGuire, “Study of exactly soluble one-dimensional N-body problem,” J. Math. Phys.,5, No. 5, 622 (1964).

    Google Scholar 

  35. I. M. Gel'fand and L. A. Dikii, “Resolvent and Hamiltonian systems,” Funkts. Anal.,11, No. 2, 11–27 (1977).

    Google Scholar 

  36. F. A. Berezin, Method of Secondary Quantization [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  37. A. S. Shvarts, Mathematical Foundations of Quantum Field Theory [in Russian], Atomizdat, Moscow (1975).

    Google Scholar 

  38. A. A. Tsvetkov, “Integrals of motion of a system of bosons with pointwise interactions,” Vestn. Mosk. Gos. Univ., Ser. Mat., Mekh., No. 4, 61–69 (1977).

    Google Scholar 

  39. F. A. Berezin, “Quantization,” Izv. Akad. Nauk SSSR, Ser. Mat.,38, No. 5, 1116–1175 (1974).

    Google Scholar 

  40. F. A. Berezin, “Wick and anti-Wick symbols of operators,” Mat. Sb.,86 (128), No. 4 (12), 578–610 (1971).

    Google Scholar 

  41. Yu. S. Tyupkin, V. A. Fateev, and A. S. Shvarts, “Connection of particle-similar solutions of classical equations with quantum particles,” Yad. Fiz.,22, No. 3, 622–631 (1975).

    Google Scholar 

Download references

Authors
  1. E. K. Sklyanin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 95, pp. 55–128, 1980.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sklyanin, E.K. Quantum version of the method of inverse scattering problem. J Math Sci 19, 1546–1596 (1982). https://doi.org/10.1007/BF01091462

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01091462

Keywords

Search

Navigation