Communications in Mathematical Physics

, Volume 359, Issue 1, pp 1–60 | Cite as

Self-Duality and Scattering Map for the Hyperbolic van Diejen Systems with Two Coupling Parameters (with an Appendix by S. Ruijsenaars)

  • Béla Gábor Pusztai


In this paper, we construct global action-angle variables for a certain two-parameter family of hyperbolic van Diejen systems. Following Ruijsenaars’ ideas on the translation invariant models, the proposed action-angle variables come from a thorough analysis of the commutation relation obeyed by the Lax matrix, whereas the proof of their canonicity is based on the study of the scattering theory. As a consequence, we show that the van Diejen system of our interest is self-dual with a factorized scattering map. Also, in an appendix by S. Ruijsenaars, a novel proof of the spectral asymptotics of certain exponential type matrix flows is presented. This result is of crucial importance in our scattering-theoretical analysis.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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
  2. 2.
    Sutherland B.: Exact results for a quantum many body problem in one dimension. Phys. Rev. A 4, 2019–2021 (1971)ADSCrossRefGoogle Scholar
  3. 3.
    Moser J.: Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16, 197–220 (1975)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Sutherland B.: Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems. World Scientific, Singapore (2004)CrossRefzbMATHGoogle Scholar
  5. 5.
    Olshanetsky M.A., Perelomov A.M.: Completely integrable Hamiltonian systems connected with semisimple Lie algebras. Invent. Math. 37, 93–108 (1976)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Olshanetsky M.A., Perelomov A.M.: Classical integrable finite-dimensional systems related to Lie algebras. Phys. Rep. 71, 313–400 (1981)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    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)ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    Ruijsenaars, S.N.M.: Finite-dimensional soliton systems. In: Kupershmidt, B. (ed.), Integrable and Superintegrable Systems. World Scientific, pp. 165–206 (1990)Google Scholar
  9. 9.
    van Diejen J.F.: Commuting difference operators with polynomial eigenfunctions. Compos. Math. 95, 183–233 (1995)MathSciNetzbMATHGoogle Scholar
  10. 10.
    van Diejen J.F.: Deformations of Calogero–Moser systems and finite Toda chains. Theor. Math. Phys. 99, 549–554 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    van Diejen J.F.: Difference Calogero–Moser systems and finite Toda chains. J. Math. Phys. 36, 1299–1323 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kazhdan D., Kostant B., Sternberg S.: Hamiltonian group actions and dynamical systems of Calogero type. Commun. Pure Appl. Math. XXXI, 481–507 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nekrasov N.: Holomorphic bundles and many-body systems. Commun. Math. Phys. 180, 587–603 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wilson G.: Collisions of Calogero–Moser particles and an adelic Grassmannian (with an Appendix by I.G. Macdonald). Invent. Math. 133, 1–41 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hurtubise J.C., Markman E.: Calogero–Moser systems and Hitchin systems. Commun. Math. Phys. 223, 533–582 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hurtubise J., Nevins T.: The geometry of Calogero–Moser systems. Ann. Inst. Fourier, Grenoble 55, 2091–2116 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Olshanetsky M.A., Perelomov A.M.: Quantum systems related to root systems, and radial parts of Laplace operators. Funct. Anal. Appl. 12, 121–128 (1978)CrossRefzbMATHGoogle Scholar
  18. 18.
    Heckmam G.J., Opdam E.M.: Root systems and hypergeometric functions I. Compos. Math. 64, 329–352 (1987)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Etingof P., Kirillov A.A. Jr.: A unified representation-theoretic approach to special functions. Funct. Anal. Appl. 28, 91–94 (1994)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Cherednik I.: Double Affine Hecke Algebras, London Mathematical Society Lecture Notes Series, vol. 319. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  21. 21.
    Etingof, P.I.: Calogero–Moser Systems and Representation Theory, European Mathematical Society (2007)Google Scholar
  22. 22.
    Avan J., Babelon O., Billey E.: The Gervais–Neveu–Felder equation and the quantum Calogero–Moser systems. Commun. Math. Phys. 178, 281–299 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Li L.C., Xu P.: A class of integrable spin Calogero–Moser systems. Commun. Math. Phys. 231, 257–286 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Etingof P., Latour F.: The Dynamical Yang–Baxter Equation, Representation Theory, and Quantum Integrable Systems. Oxford University Press, Oxford (2005)zbMATHGoogle Scholar
  25. 25.
    D’Hoker E., Phong D.H.: Seiberg–Witten theory and Calogero–Moser systems. Prog. Theor. Phys. Suppl. 135, 75–93 (1999)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Blom J., Langmann E.: Finding and solving Calogero–Moser type systems using Yang–Mills gauge theories. Nucl. Phys. B 563, 506–532 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Gorsky A., Zabrodin A., Zotov A.: Spectrum of quantum transfer matrices via classical many-body systems. JHEP 01, 070 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tsuboi Z., Zabrodin A., Zotov A.: Supersymmetric quantum spin chains and classical integrable systems. JHEP 05, 086 (2015)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Beketov M., Liashyk A., Zabrodin A., Zotov A.: Trigonometric version of quantum-classical duality in integrable systems. Nucl. Phys. B 903, 150–163 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Bogomolny E., Giraud O., Schmit C.: Random matrix ensembles associated with Lax matrices. Phys. Rev. Lett. 103, 054103 (2009)ADSCrossRefGoogle Scholar
  33. 33.
    Bogomolny E., Giraud O., Schmit C.: Integrable random matrix ensembles. Nonlinearity 24, 3179–3213 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Fyodorov Y.V., Giraud O.: High values of disorder-generated multifractals and logarithmically correlated processes. Chaos, Solitons & Fractals 74, 15–26 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Aminov G., Arthamonov S., Smirnov A., Zotov A.: Rational top and its classical R-matrix. J. Phys. A: Math. Theor. 47, 305207 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Levin A., Olshanetsky M., Zotov A.: Relativistic classical integrable tops and quantum R-matrices. JHEP 07, 012 (2014)ADSCrossRefGoogle Scholar
  37. 37.
    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)ADSCrossRefzbMATHGoogle Scholar
  38. 38.
    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)zbMATHGoogle Scholar
  39. 39.
    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)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Babelon O., Bernard D.: The sine-Gordon solitons as an N-body problem. Phys. Lett. B 317, 363–368 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Nekrasov, N.: Infinite-dimensional algebras, many-body systems and gauge theories. In: Morozov, A.Yu, Olshanetsky, M.A. (eds.), Moscow Seminar in Mathematical Physics, AMS Transl. Ser. 2, vol. 191, American Mathematical Society, Providence, pp. 263–299 (1999)Google Scholar
  42. 42.
    Fock V., Gorsky A., Nekrasov N., Rubtsov V.: Duality in integrable systems and gauge theories. JHEP 07, 028 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Arutyunov G.E., Frolov S.A., Medvedev P.B.: Elliptic Ruijsenaars–Schneider model via the Poisson reduction of the affine Heisenberg double. J. Phys. A 30, 5051–5063 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Arutyunov G.E., Frolov S.A., Medvedev P.B.: Elliptic Ruijsenaars–Schneider model from the cotangent bundle over the two-dimensional current group. J. Math. Phys. 38, 5682–5689 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Fehér L., Ayadi V.: Trigonometric Sutherland systems and their Ruijsenaars duals from symplectic reduction. J. Math. Phys. 51, 103511 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Fehér L., Klimčík C.: Poisson–Lie interpretation of trigonometric Ruijsenaars duality. Commun. Math. Phys. 301, 55–104 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Fehér L., Klimčík C.: Self-duality of the compactified Ruijsenaars–Schneider system from quasi-Hamiltonian reduction. Nucl. Phys. B 860, 464–515 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Fehér L., Kluck T.J.: New compact forms of the trigonometric Ruijsenaars–Schneider system. Nucl. Phys. B 882, 97–127 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Fock, V.V., Rosly, A.A.: Poisson structure on moduli of flat connections on Riemann surfaces and the r-matrix. In: Morozov, A.Yu., Olshanetsky, M.A. (eds.), Moscow Seminar in Mathematical Physics, AMS Transl. Ser. 2, vol. 191, American Mathematical Society, Providence, pp. 67–86 (1999)Google Scholar
  51. 51.
    Oblomkov A.: Double affine Hecke algebras and Calogero–Moser spaces. Represent. Theory 8, 243–266 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Chalykh, O., Fairon, M.: Multiplicative quiver varieties and generalised Ruijsenaars–Schneider models. arXiv:1704.05814 [math-ph]
  53. 53.
    Chen K., Hou B.Y.: The D n Ruijsenaars–Schneider model. J. Phys. A 34, 7579–7589 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    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)ADSCrossRefzbMATHGoogle Scholar
  56. 56.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    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)ADSCrossRefzbMATHGoogle Scholar
  58. 58.
    Marshall I.: A new model in the Calogero–Ruijsenaars family. Commun. Math. Phys. 338, 563–587 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Fehér L., Görbe T.F.: The full phase space of a model in the Calogero–Ruijsenaars family. J. Geom. Phys. 115, 139–149 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Fehér L., Marshall I.: The action-angle dual of an integrable Hamiltonian system of Ruijsenaars–Schneider–van Diejen type. J. Phys. A: Math. Theor. 50, 314004 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Ruijsenaars S.N.M.: The classical hyperbolic Askey–Wilson dynamics without bound states. Theor. Math. Phys. 154, 418–432 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Pusztai B.G., Görbe T.F.: Lax representation of the hyperbolic van Diejen dynamics with two coupling parameters. Commun. Math. Phys. 354, 829–864 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Knapp A.W.: Lie Groups Beyond an Introduction, Progress in Mathematics, vol. 140. Birkhäuser, Boston (2002)Google Scholar
  64. 64.
    Pusztai B.G.: On the scattering theory of the classical hyperbolic C n Sutherland model. J. Phys. A: Math. Theor. 44, 155306 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Prasolov, V.V., Problems and Theorems in Linear Algebra, American Mathematical Society, Providence (1994)Google Scholar
  66. 66.
    Ruijsenaars S.N.M.: Complete integrability of relativistic Calogero–Moser systems and elliptic function identities. Commun. Math. Phys. 110, 191–213 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Lee J.M.: Introduction to Smooth Manifolds, 2nd edn., Graduate Texts in Mathematics, vol. 218. Springer, New York (2013)Google Scholar
  68. 68.
    Hartman, P.: Ordinary Differential Equations, 2nd edn., SIAM, Philadelphia (2002)Google Scholar
  69. 69.
    Folland G.B.: Real Analysis: Modern Techniques and Their Applications, 2nd edn. Wiley-Interscience, John Wiley & Sons, New York (1999)zbMATHGoogle Scholar
  70. 70.
    Kulish P.P.: Factorization of the classical and the quantum S matrix and conservation laws. Theor. Math. Phys. 26, 132–137 (1976)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Moser, J.: The scattering problem for some particle systems on the line. In: Lecture Notes in Mathematics, vol. 597, Springer, New York, pp. 441–463 (1977)Google Scholar
  72. 72.
    Saleur M., Skorik S., Warner N.P.: The boundary sine-Gordon theory: classical and semi-classical analysis. Nucl. Phys. B 441, 421–436 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    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
  74. 74.
    Reed M., Simon B.: Methods of Modern Mathematical Physics. IV: Analysis of Operators.. Academic Press, New York (1978)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Bolyai InstituteUniversity of SzegedSzegedHungary
  2. 2.MTA Lendület Holographic QFT Group, Wigner RCPBudapest 114Hungary

Personalised recommendations