Communications in Mathematical Physics

, Volume 354, Issue 3, pp 829–864 | Cite as

Lax Representation of the Hyperbolic van Diejen Dynamics with Two Coupling Parameters



In this paper, we construct a Lax pair for the classical hyperbolic van Diejen system with two independent coupling parameters. Built upon this construction, we show that the dynamics can be solved by a projection method, which in turn allows us to initiate the study of the scattering properties. As a consequence, we prove the equivalence between the first integrals provided by the eigenvalues of the Lax matrix and the family of van Diejen’s commuting Hamiltonians. Also, at the end of the paper, we propose a candidate for the Lax matrix of the hyperbolic van Diejen system with three independent coupling constants.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    van Diejen J.F.: Commuting difference operators with polynomial eigenfunctions. Compos. Math. 95, 183–233 (1995)MathSciNetMATHGoogle Scholar
  2. 2.
    van Diejen J.F.: Deformations of Calogero–Moser systems. Theor. Math. Phys. 99, 549–554 (1994)CrossRefMATHGoogle Scholar
  3. 3.
    van Diejen J.F.: Difference Calogero–Moser systems and finite Toda chains. J. Math. Phys. 36, 1299–1323 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ruijsenaars S.N.M., Schneider H.: A new class of integrable models and its relation to solitons. Ann. Phys. (N.Y.) 170, 370–405 (1986)ADSCrossRefMATHGoogle Scholar
  5. 5.
    Ruijsenaars S.N.M.: Complete integrability of relativistic Calogero–Moser systems and elliptic function identities. Commun. Math. Phys. 110, 191–213 (1987)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Calogero F.: Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials. J. Math. Phys. 12, 419–436 (1971)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Sutherland B.: Exact results for a quantum many body problem in one dimension. Phys. Rev. A 4, 2019–2021 (1971)ADSCrossRefGoogle Scholar
  8. 8.
    Moser J.: Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16, 197–220 (1975)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Olshanetsky M.A., Perelomov A.M.: Completely integrable Hamiltonian systems connected with semisimple Lie algebras. Invent. Math. 37, 93–108 (1976)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Pusztai B.G.: Action-angle duality between the C n-type hyperbolic Sutherland and the rational Ruijsenaars–Schneider–van Diejen models. Nucl. Phys. B 853, 139–173 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Pusztai B.G.: The hyperbolic BC n Sutherland and the rational BC n Ruijsenaars–Schneider–van Diejen models: Lax matrices and duality. Nucl. Phys. B 856, 528–551 (2012)ADSCrossRefMATHGoogle Scholar
  12. 12.
    Pusztai B.G.: Scattering theory of the hyperbolic BC n Sutherland and the rational BC n Ruijsenaars–Schneider–van Diejen models. Nucl. Phys. B 874, 647–662 (2013)ADSCrossRefMATHGoogle Scholar
  13. 13.
    Fehér L., Görbe T.F.: Duality between the trigonometric BC n Sutherland system and a completed rational Ruijsenaars–Schneider–van Diejen system. J. Math. Phys. 55, 102704 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Görbe T.F., Fehér L.: Equivalence of two sets of Hamiltonians associated with the rational BC n Ruijsenaars–Schneider–van Diejen system. Phys. Lett. A 379, 2685–2689 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Pusztai B.G.: On the classical r-matrix structure of the rational BC n Ruijsenaars–Schneider–van Diejen system. Nucl. Phys. B 900, 115–146 (2015)ADSCrossRefMATHGoogle Scholar
  16. 16.
    Knapp, A.W.: Lie Groups Beyond an Introduction, Progress in Mathematics, vol. 140, Birkhäuser, Boston (2002)Google Scholar
  17. 17.
    Ruijsenaars S.N.M.: Action-angle maps and scattering theory for some finite dimensional integrable systems I. The pure soliton case. Commun. Math. Phys. 115, 127–165 (1988)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Abraham R., Marsden J.E.: Foundations of Mechanics, 2 edn. Addison Wesley, Boston (1985)Google Scholar
  19. 19.
    Olshanetsky M.A., Perelomov A.M.: Classical integrable finite-dimensional systems related to Lie algebras. Phys. Rep. 71, 313–400 (1981)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Kazhdan D., Kostant B., Sternberg S.: Hamiltonian group actions and dynamical systems of Calogero type. Commun. Pure Appl. Math. XXXI, 481–507 (1978)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Fehér L., Pusztai B.G.: Spin Calogero models associated with Riemannian symmetric spaces of negative curvature. Nucl. Phys. B 751, 436–458 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Fehér L., Pusztai B.G.: A class of Calogero type reductions of free motion on a simple Lie group. Lett. Math. Phys. 79, 263–277 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Fehér L., Klimčík C.: On the duality between the hyperbolic Sutherland and the rational Ruijsenaars–Schneider models. J. Phys. A: Math. Theor. 42, 185202 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Ruijsenaars, S.N.M.: Finite-dimensional soliton systems. In: Kupershmidt, B. (ed.), Integrable and superintegrable systems, World Scientific, pp. 165-206 (1990)Google Scholar
  25. 25.
    Kulish P.P.: Factorization of the classical and the quantum S matrix and conservation laws. Theor. Math. Phys. 26, 132–137 (1976)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Moser, J.: The scattering problem for some particle systems on the line. In: Lecture Notes in Mathematics, vol. 597, pp. 441–463. Springer, New York (1977)Google Scholar
  27. 27.
    Babelon O., Bernard D.: The sine-Gordon solitons as an N-body problem. Phys. Lett. B 317, 363–368 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Ruijsenaars S.N.M.: Action-angle maps and scattering theory for some finite dimensional integrable systems II. Solitons, antisolitons and their bound states. Publ. RIMS 30, 865–1008 (1994)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Ruijsenaars S.N.M.: Action-angle maps and scattering theory for some finite dimensional integrable systems III. Sutherland type systems and their duals. Publ. RIMS 31, 247–353 (1995)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Saleur H., Skorik S., Warner N.P.: The boundary sine-Gordon theory: classical and semi-classical analysis. Nucl. Phys. B 441, 421–436 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Kapustin A., Skorik S.: On the non-relativistic limit of the quantum sine-Gordon model with integrable boundary condition. Phys. Lett. A 196, 47–51 (1994)ADSCrossRefGoogle Scholar
  32. 32.
    Mukhin E., Tarasov V., Varchenko A.: Gaudin Hamiltonians generate the Bethe algebra of a tensor power of the vector representation of \({\mathfrak{gl}_N}\) . St. Petersburg Math. J. 22, 463–472 (2011)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Alexandrov A., Leurent S., Tsuboi Z., Zabrodin A.: The master T-operator for the Gaudin model and the KP hierarchy. Nucl. Phys. B 883, 173–223 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Gorsky A., Zabrodin A., Zotov A.: Spectrum of quantum transfer matrices via classical many-body systems. JHEP 01, 070 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Tsuboi Z., Zabrodin A., Zotov A.: Supersymmetric quantum spin chains and classical integrable systems. JHEP 05, 086 (2015)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Beketov M., Liashyk A., Zabrodin A., Zotov A.: Trigonometric version of quantum-classical duality in integrable systems. Nucl. Phys. B 903, 150–163 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Aminov G., Arthamonov S., Smirnov A., Zotov A.: Rational top and its classical R-matrix. J. Phys. A: Math. Theor. 47, 305207 (2014)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Levin A., Olshanetsky M., Zotov A.: Relativistic classical integrable tops and quantum R-matrices. JHEP 07, 012 (2014)ADSCrossRefGoogle Scholar
  39. 39.
    Fehér L., Ayadi V.: Trigonometric Sutherland systems and their Ruijsenaars duals from symplectic reduction. J. Math. Phys. 51, 103511 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Fehér L., Klimčík C.: Poisson–Lie interpretation of trigonometric Ruijsenaars duality. Commun. Math. Phys. 301, 55–104 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Fehér L., Klimčík C.: Self-duality of the compactified Ruijsenaars–Schneider system from quasi-Hamiltonian reductions. Nucl. Phys. B 860, 464–515 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Bogomolny E., Giraud O., Schmit C.: Random matrix ensembles associated with Lax matrices. Phys. Rev. Lett. 103, 054103 (2009)ADSCrossRefGoogle Scholar
  43. 43.
    Bogomolny E., Giraud O., Schmit C.: Integrable random matrix ensembles. Nonlinearity 24, 3179–3213 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Fyodorov Y.V., Giraud O.: High values of disorder-generated multifractals and logarithmically correlated processes. Chaos Solitons Fract. 74, 15–26 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Krichever I.M.: Elliptic solutions of the Kadomtsev–Petviashvili equation and integrable systems of particles. Funct. Anal. Appl. 14, 282–290 (1980)CrossRefMATHGoogle Scholar
  46. 46.
    Babelon O., Bernard D., Talon M.: Introduction to Classical Integrable Systems. Cambridge University Press, Cambridge (2003)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Bolyai InstituteUniversity of SzegedSzegedHungary
  2. 2.MTA Lendület Holographic QFT GroupWigner RCPBudapest 114Hungary
  3. 3.Department of Theoretical PhysicsUniversity of SzegedSzegedHungary

Personalised recommendations