Symmetric and non-symmetric bases of quantum integrable particle systems with long-range interactions

Journal of Statistical Physics

Volumn 102, Issue 3-4, 2001, Pages 1049-1064

  Wadati, Miki ^a   Nishino, Akinori ^a   Ujino, Hideaki ^b   Komori, Yasushi ^a  

^a UNIVERSITY OF TOKYO   (Japan)

^b GUNMA NATIONAL COLLEGE OF TECHNOLOGY   (Japan)

Author keywords

Calogero model; Dunkl Cherednik operator; Hi Jack polynomial; Long range interaction; Multivariable Hermite polynomial; Quantum integrable system; Rodrigues formula; Weyl group

Indexed keywords

EID: 0035534211     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1004875625099     Document Type: Article

References (34)

1. T. H. Baker and P. J. Forrester, The Calogero Sutherland model and polynomials with prescribed symmetry, Nucl. Phys. B 492:682-716 (1997).
2. T. H. Baker and P. J. Forrester, Nonsymmetric Jack polynomials and integral kernels, Duke Math. J. 95:1-50 (1998).
3. F. Calogero, Solution of the one-dimensional N-body problem with quadratic and or inversely quadratic pair potentials, J. Math. Phys. 12:419-436 (1971).
4. I. Cherednik, A unification of the Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Invent. Math. 106:411-432 (1991).
5. I. Cherednik, Double affine Hecke algebras and Macdonald's conjectures, Ann. Math. 95:191-216 (1995).
6. J. F. van Diejen, Confluent hypergeometric orthogonal polynomials related to the rational Calogero system with harmonic confinement. Commun. Math. Phys. 188:467-497 (1997).
7. C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Am. Math. Soc. 311:167-183 (1989).
8. K. Hikami and M. Wadati, Infinite symmetry of the spin systems with inverse square interactions, J. Phys. Soc. Jpn. 62:4203-4217 (1993).
9. H. Jack, A class of symmetric polynomials with a parameter, Proc. Roy. Soc. Edinburg Sect. A 69:1-18 (1970).
10. S. Kakei, Intertwining operators for a degenerate double affine Hecke algebra and multivariable orthogonal polynomials, J. Math. Phys. 39:4993-5006 (1998).
11. F. Knop and S. Sahi, A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128:9-22 (1997).
12. Y. Komori and K. Hikami, Affine R-matrix and the generalized elliptic Ruijsenaars models, Lett. Math. Phys. 43:335-346 (1998).
13. Y. Komori, Theta functions associated with the affine root systems and the elliptic Ruijsenaars operators, preprint (math. QA/9910003).
14. L. Lapointe and L. Vinet, A Rodrigues formula for the Jack polynomials and the Macdonald-Stanley conjecture, Int. Math. Res. Not. 9:419-424 (1995).
15. M. Lassalle, Polynômes de Hermite généralisés, C.R. Acad. Sci. Paris t. Séries I 313:579-582 (1991).
16. I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed. (Clarendon Press, Oxford, 1995).
17. A. Nishino, Y. Komori, H. Ujino, and M. Wadati, Symmetrization of the nonsymmetric Macdonald polynomials and Macdonald's inner product identities, preprint.
18. A. Nishino, H. Ujino, and M. Wadati, Rodrigues formula for the nonsymmetric multivariable Laguerre polynomial, J. Phys. Soc. Jpn. 68:797-802 (1999).
19. A. Nishino, H. Ujino, and M. Wadati, An algebraic approach for the non-symmetric Macdonald polynomial, Nucl. Phys. B 558:589-603 (1999).
20. N-type root system, Nucl. Phys. B 571:632-648 (2000).
21. A. Nishino and M. Wadati, Bosonic and fermionic eigenstates for generalized Sutherland models, J. Phys. A: Math. Gen. 33:1351-1361 (2000).
22. M. A. Olshanetsky and A. M. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Rep. 94:313-404 (1983).
23. A. P. Polychronakos, Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69:703-705 (1992).
24. B. Sutherland, Exact results for a quantum many-body problem in one dimension I, Phys. Rev. A 4:2019-2021 (1971).
25. B. Sutherland, Exact results for a quantum many-body problem in one dimension II, Phys. Rev. A 5:1372-1376 (1972).
26. H. Ujino and M. Wadati, The quantum Calogero model and H'-Algebra. J. Phys. Soc. Jpn. 63:3585-3597 (1994).
27. H. Ujino and M. Wadati, Algebraic approach to the eigenstates of the quantum Calogero model confined in an external harmonic well. Chaos Solitons & Fractals 5:109-118 (1995).
28. H. Ujino and M. Wadati, The SU(ν) quantum Calogero model and H' , -Algebra with SU(ν) symmetry, J. Phys. Soc. Jpn. 64:39-56 (1995).
29. H. Ujino and M. Wadati, Algebraic construction of the eigenstates for the second conserved operator of the quantum Calogero model, J. Phys. Soc. Jpn. 65:653-656 (1996).
30. H. Ujino and M. Wadati, Rodrigues formula for Hi-Jack symmetric polynomials associated with the quantum Calogero model, J. Phys. Soc. Jpn. 65:2423-2439 (1996).
31. H. Ujino and M. Wadati, Orthogonality of the Hi Jack polynomials associated with the Calogero model, J. Phys. Soc. Jpn. 66:345-350 (1997).
32. H. Ujino and M. Wadati, Rodrigues formula for the nonsymmetric multivariable Hermite polynomial, J. Phys. Soc. Jpn. 68:391-395 (1999).
33. H. Ujino and A. Nishino, Rodrigues formulas for nonsymmetric multivariable polynomials associated with quantum integrable systems of Calogero Sutherland type, submitted to Proc. of International Workshop on "Special Functions Asymptotics, Harmonic Analysis and Mathematical Physics." June 21-25 (1999), City Univ. Hong Kong, Hong Kong, China.
34. M. Wadati, T. Deguchi, and Y. Akutsu, Exactly solvable models and knot theory, Phys. Rep. 180:247-332 (1989).