Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 013, 14 pages      nlin.SI/0701040      https://doi.org/10.3842/SIGMA.2007.013
Contribution to the Vadim Kuznetsov Memorial Issue

Relativistic Toda Chain with Boundary Interaction at Root of Unity

Nikolai Iorgov a, Vladimir Roubtsov b, c, Vitaly Shadura a and Yuri Tykhyy a
a) Bogolyubov Institute for Theoretical Physics, 14b Metrolohichna Str., Kyiv, 03143 Ukraine
b) LAREMA, Dépt. de Math. Université d'Angers, 2 bd. Lavoisier, 49045, Angers, France
c) ITEP, Moscow, 25 B. Cheremushkinskaja, 117259, Moscow, Russia

Received November 15, 2006, in final form January 03, 2007; Published online January 19, 2007

Abstract
We apply the Separation of Variables method to obtain eigenvectors of commuting Hamiltonians in the quantum relativistic Toda chain at a root of unity with boundary interaction.

Key words: quantum integrable model with boundary interaction; quantum relativistic Toda chain.

pdf (265 kb)   ps (191 kb)   tex (16 kb)

References

  1. Sklyanin E., Separation of variable. New trends, Prog. Theoret. Phys. Suppl. 118 (1995), 35-60, solv-int/9504001.
  2. Kharchev S., Lebedev D., Integral representation for the eigenfunctions of quantum periodic Toda chain, Lett. Math. Phys. 50 (1999), 53-77, hep-th/9910265.
  3. Kharchev S., Lebedev D., Semenov-Tian-Shansky M., Unitary representations of Uq(sl(2,R)), the modular double, and the multiparticle q-deformed Toda chains, Comm. Math. Phys. 225 (2002), 573-609, hep-th/0102180.
  4. Kuznetsov V., Separation of variables for the Dn type periodic Toda lattice, J. Phys. A: Math. Gen. 30 (1997), 2127-2138, solv-int/9701009.
  5. Iorgov N., Shadura V., Wave functions of the Toda chain with boundary interactions, Theor. Math. Phys. 142 (2005), 289-305, nlin.SI/0411002.
  6. Sklyanin E., Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen. 21 (1988), 2375-2389.
  7. von Gehlen G., Iorgov N., Pakuliak S., Shadura V., Baxter-Bazhanov-Stoganov model: separation of variables and Baxter equation, J. Phys. A: Math. Gen. 39 (2006), 7257-7282, nlin.SI/0603028.
  8. Iorgov N., Eigenvectors of open Bazhanov-Stroganov quantum chain, SIGMA 2 (2006), 019, 10 pages, nlin.SI/0602010.
  9. Bazhanov V.V., Stroganov Yu.G., Chiral Potts model as a descendant of the six-vertex model, J. Statist. Phys. 59 (1990), 799-817.
  10. Korepanov I.G., Hidden symmetries in the 6-vertex model of statistical physics, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 215 (1994), 163-177 (English transl.: J. Math. Sci. (New York) 85 (1997), 1661-1670, hep-th/9410066).
  11. Pakuliak S., Sergeev S., Quantum relativistic Toda chain at root of unity: isospectrality, modified Q-operator and functional Bethe ansatz, Int. J. Math. Math. Sci. 31 (2002), 513-554, nlin.SI/0205037.
  12. Enriquez B., Rubtsov V., Commuting families in skew fields and quantization of Beauville's fibrations, Duke Math. J. 82 (2003), 197-219, math.AG/0112276.
  13. Sergeev S., Coefficient matrices of a quantum discrete auxiliary linear problem, Zap. Nauchn. Sem. POMI 269 (2000), no. 16, 292-307 (in Russian).
  14. Babelon O., Talon M., Riemann surfaces, separation of variables and classical and quantum integrability, Phys. Lett. A 312 (2003), 71-77, hep-th/0209071.
  15. Kuznetsov V.B., Tsiganov A.V., Infinite series of Lie algebras and boundary conditions for integrable systems, J. Sov. Math. 59 (1992), 1085-1092.
  16. Kuznetsov V.B., Tsiganov A.V., Separation of variables for the quantum relativistic Toda lattices, Report 94-07, Mathematical Preprint Series, University of Amsterdam, 1994, hep-th/9402111.
  17. Kuznetsov V.B., Jorgensen M.F., Christiansen P.L., New boundary conditions for integrable lattices, J. Phys. A: Math. Gen. 28 (1995), 4639-4654, hep-th/9503168.
  18. de Vega H.J., Gonzalez-Ruiz A., Boundary K-matrices for the XYZ, XXZ and XXX spin chains, J. Phys. A: Math. Gen. 28 (1994), 6129-6141.
  19. Bazhanov V.V., Baxter R.J., Star-triangle relation for a three dimensional model, J. Statist. Phys. 71 (1993), 839-864, hep-th/9212050.
  20. Bugrij A.I., Iorgov N.Z., Shadura V.N., Alternative method of calculating the eigenvalues of the transfer matrix of the t2 model for N = 2, JETP Lett. 119 (2005), no. 2, 311-315.

Previous article   Next article   Contents of Volume 3 (2007)