Theoretical and Mathematical Physics

, Volume 150, Issue 3, pp 315–331 | Cite as

Transition function for the Toda chain

  • A. V. Silantyev


We use the method of Λ-operators developed by Derkachov, Korchemsky, and Manashov to derive eigenfunctions for the open Toda chain. Using the diagram technique developed for these Λ-operators, we reproduce the Sklyanin measure and study the properties of the Λ-operators. This approach to the open Toda chain eigenfunctions reproduces the Gauss-Givental representation for these eigenfunctions.


Toda chain separation of variables Q-operators 


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  1. 1.
    S. Derkachov, G. Korchemsky, and A. Manashov, JHEP, 07, 047 (2003).CrossRefMathSciNetGoogle Scholar
  2. 2.
    E. K. Sklyanin, “The quantum Toda chain,” in: Nonlinear Equations in Classical and Quantum Field Theory (Lect. Notes Phys., Vol. 226, N. Sanchez and H. DeVega, eds.), Springer, Berlin (1985), pp. 196–233.Google Scholar
  3. 3.
    S. Kharchev and D. Lebedev, Lett. Math. Phys., 50, 53–77 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    V. Pasquier and M. Gaudin, J. Phys. A, 25, 5243–5252 (1992).zbMATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    M. Gutzwiller, Ann. Phys., 133, 304–331 (1981).CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    A. Gerasimov, S. Kharchev, D. Lebedev, and S. Oblezin, “On a Gauss-Givental representation of quantum Toda chain wave function,” math.RT/0505310 (2005).Google Scholar
  7. 7.
    A. Givental, “Stationary phase integrals, quantum Toda lattices, flag manifolds, and the mirror conjecture,” in: Topics in Singularity Theory: V. I. Arnold’s 60th Anniversary Collection (Am. Math. Soc. Transl., Ser. 2, Vol. 180, A. Khovanskii, A. Varchenko, and V. Vassiliev, eds.), Amer. Math. Soc., Providence, R. I. (1997), pp. 103–115.Google Scholar
  8. 8.
    R. Baxter, Exactly Solved Models in Statistical Mechanics, Acad. Press, London (1982).zbMATHGoogle Scholar
  9. 9.
    H. Bateman and A. Erdélyi, eds., Higher Transcendental Functions (Based on notes left by H. Bateman), Vol. 2, McGraw-Hill, New York (1953).Google Scholar
  10. 10.
    S. Kharchev and D. Lebedev, JETP Letters, 71, 235–238 (2000).CrossRefADSGoogle Scholar
  11. 11.
    S. Kharchev and D. Lebedev, J. Phys. A, 34, 2247–2258 (2001).zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    I. M. Gelfand and N. Ya. Vilenkin, Generalized Functions [in Russian], Vol. 4, Some Applications of Harmonic Analysis: Rigged Hilbert Spaces, Fizmatgiz, Moscow (1961); English transl.: Generalized Functions, Vol. 4, Applications of Harmonic Analysis, Acad. Press, New York (1964).Google Scholar
  13. 13.
    V. S. Vladimirov, Generalized Functions in Mathematical Physics [in Russian], Nauka, Moscow (1976); English transl., Mir, Moscow (1979).Google Scholar
  14. 14.
    O. Babelon, Lett. Math. Phys., 65, 229–246 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    A. Gerasimov, S. Kharchev, and D. Lebedev, Internat. Math. Res. Notices, 17, 823–854 (2004).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • A. V. Silantyev
    • 1
    • 2
  1. 1.Joint Institute for Nuclear ResearchDubna, Moscow OblastRussia
  2. 2.Département de MathématiquesUniversité d’AngersAngersFrance

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