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


SIGMA 9 (2013), 077, 12 pages      arXiv:1312.1028      https://doi.org/10.3842/SIGMA.2013.077
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

Boundary Interactions for the Semi-Infinite q-Boson System and Hyperoctahedral Hall-Littlewood Polynomials

Jan Felipe van Diejen and Erdal Emsiz
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile

Received September 27, 2013, in final form November 26, 2013; Published online December 04, 2013

Abstract
We present a semi-infinite q-boson system endowed with a four-parameter boundary interaction. The n-particle Hamiltonian is diagonalized by generalized Hall-Littlewood polynomials with hyperoctahedral symmetry that arise as a degeneration of the Macdonald-Koornwinder polynomials and were recently studied in detail by Venkateswaran.

Key words: Hall-Littlewood functions; q-bosons; boundary fields; hyperoctahedral symmetry.

pdf (351 kb)   tex (16 kb)

References

  1. Bogoliubov N.M., Izergin A.G., Kitanine N.A., Correlation functions for a strongly correlated boson system, Nuclear Phys. B 516 (1998), 501-528, solv-int/9710002.
  2. Borodin A., Corwin I., Petrov L., Sasamoto T., Spectral theory for the q-boson particle system, arXiv:1308.3475.
  3. van Diejen J.F., Properties of some families of hypergeometric orthogonal polynomials in several variables, Trans. Amer. Math. Soc. 351 (1999), 233-270, q-alg/9604004.
  4. van Diejen J.F., Emsiz E., Diagonalization of the infinite q-boson system, arXiv:1308.2237.
  5. van Diejen J.F., Emsiz E., The semi-infinite q-boson system with boundary interaction, Lett. Math. Phys., to appear, arXiv:1308.2242.
  6. Klimyk A., Schmüdgen K., Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.
  7. Koornwinder T.H., Askey-Wilson polynomials for root systems of type BC, in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, Amer. Math. Soc., Providence, RI, 1992, 189-204.
  8. Korff C., Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra, Comm. Math. Phys. 318 (2013), 173-246, arXiv:1110.6356.
  9. Macdonald I.G., Orthogonal polynomials associated with root systems, Sém. Lothar. Combin. 45 (2000), Art. B45a, 40 pages, math.QA/0011046.
  10. Macdonald I.G., Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge, 2003.
  11. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
  12. Povolotsky A.M., On integrability of zero-range chipping models with factorized steady state, J. Phys. A: Math. Theor. 46 (2013), 465205, 25 pages, arXiv:1308.3250.
  13. Sasamoto T., Wadati M., Exact results for one-dimensional totally asymmetric diffusion models, J. Phys. A: Math. Gen. 31 (1998), 6057-6071.
  14. Takeyama Y., A discrete analogue of periodic delta Bose gas and affine Hecke algebra, arXiv:1209.2758.
  15. Tsilevich N.V., The quantum inverse scattering problem method for the q-boson model and symmetric functions, Funct. Anal. Appl. 40 (2006), 207-217, math-ph/0510073.
  16. Venkateswaran V., Symmetric and nonsymmetric Hall-Littlewood polynomials of type BC, arXiv:1209.2933.

Previous article  Next article   Contents of Volume 9 (2013)