Abstract
These lecture notes are intended as an introduction to the theory of rational Dunkl operators and the associated special functions, with an emphasis on positivity and asymptotics. We start with an outline of the general concepts: Dunkl operators, the intertwining operator, the Dunkl kernel and the Dunkl transform. We point out the connection with integrable particle systems of Calogero-Moser-Sutherland type, and discuss some systems of orthogonal polynomials associated with them. A major part is devoted to positivity results for the intertwining operator and the Dunkl kernel, the Dunkl-type heat semigroup, and related probabilistic aspects. The notes conclude with recent results on the asymptotics of the Dunkl kernel.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A., Pocketbook of Mathematical Functions. Verlag Harri Deutsch, Frankfurt/Main, 1984.
Baker, T.H., Forrester, P.J., The Calogero-Sutherland model and generalized classical polynomials. Comm. Math. Phys. 188 (1997), 175–216.
___, The Calogero-Sutherland model and polynomials with prescribed symmetry. Nucl. Phys. B 492 (1997), 682–716.
___, Non-symmetric Jack polynomials and integral kernels. Duke Math. J. 95 (1998), 1–50.
Brink, L., Hansson, T.H., Konstein, S., Vasiliev, M.A., The Calogero model-anyonic representation, fermionic extension and supersymmetry. Nucl. Phys. B 401 (1993), 591–612.
Calogero, F., Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials. J. Math. Phys. 12 (1971), 419–436.
Chihara, T.S., An Introduction to Orthogonal Polynomials. Gordon and Breach, 1978.
Davies, E.B.: Spectral Theory and Differential Operators. Cambridge University Press, 1995.
van Diejen, J.F., Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement. Comm. Math. Phys. 188 (1997), 467–497.
van Diejen, J.F., Vinet, L. Calogero-Sutherland-Moser Models. CRM Series in Mathematical Physics, Springer-Verlag, 2000.
Dunkl, C.F., Reflection groups and orthogonal polynomials on the sphere. Math. Z. 197 (1988), 33–60.
___, Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311 (1989), 167–183.
___, Operators commuting with Coxeter group actions on polynomials. In: Stanton, D. (ed.), Invariant Theory and Tableaux, Springer, 1990, pp. 107–117.
___, Integral kernels with reflection group invariance. Canad. J. Math. 43 (1991), 1213–1227.
___, Hankel transforms associated to finite reflection groups. In: Proc. of the special session on hypergeometric functions on domains of positivity, Jack polynomials and applications. Proceedings, Tampa 1991, Contemp. Math. 138 (1992), pp. 123–138.
___, Intertwining operators associated to the group S3. Trans. Amer. Math. Soc. 347 (1995), 3347–3374.
___, Symmetric Functions and BN-invariant spherical harmonics. Preprint; math.CA/0207122.
Dunkl, C.F., de Jeu M.F.E., Opdam, E.M., Singular polynomials for finite reflection groups. Trans. Amer. Math. Soc. 346 (1994), 237–256.
Dunkl, C.F., Xu, Yuan, Orthogonal Polynomials of Several Variables; Cambridge Univ. Press, 2001.
Eastham, M.S.P., The Asymptotic Solution of Linear Differential Systems: Applications of the Levinson Theorem. Clarendon Press, Oxford 1989.
Fell, J.M.G., Doran, R.S., Representations of-Algebras, Locally Compact Groups, and Banach—Algebraic Bundles, Vol. 1. Academic Press, 1988.
Graham, C., McGehee, O.C., Essays in Commutative Harmonic Analysis, Springer Grundlehren 238, Springer-Verlag, New York 1979.
Grove, L.C., Benson, C.T., Finite Reflection Groups; Second edition. Springer, 1985.
Ha, Z.N.C., Exact dynamical correlation functions of the Calogero-Sutherland model and one dimensional fractional statistics in one dimension: View from an exactly solvable model. Nucl. Phys. B 435 (1995), 604–636.
Haldane, D., Physics of the ideal fermion gas: Spinons and quantum symmetries of the integrable Haldane-Shastry spin chain. In: A. Okiji, N. Kamakani (eds.), Correlation effects in low-dimensional electron systems. Springer, 1995, pp. 3–20.
Heckman, G.J., A remark on the Dunkl differential-difference operators. In: Barker, W., Sally, P. (eds.) Harmonic analysis on reductive groups. Progress in Math. 101, Birkhäuser, 1991. pp. 181–191.
___, Dunkl operators. Séminaire Bourbaki 828, 1996-97; Astérisque 245 (1997), 223–246.
Helgason, S., Groups and Geometric Analysis. American Mathematical Society, 1984.
Humphreys, J.E., Reflection Groups and Coxeter Groups. Cambridge University Press, 1990.
de Jeu, M.F.E., The Dunkl transform. Invent. Math. 113 (1993), 147–162.
___, Dunkl operators. Thesis, University of Leiden, 1994.
___, Subspaces with equal closure. Preprint. math.CA/0111015.
Kakei, S., Common algebraic structure for the Calogero-Sutherland models. J. Phys. A 29 (1996), L619–L624.
Kallenberg, O., Foundations of Modern Probability. Springer-Verlag, 1997.
Knop, F., Sahi, S., A recursion and combinatorial formula for Jack polynomials. Invent. Math. 128 (1997), 9–22.
Lapointe L., Vinet, L., Exact operator solution of the Calogero-Sutherland model. Comm. Math. Phys. 178 (1996), 425–452.
Lassalle, M., Polynômes de Laguerre généralisés. C.R. Acad. Sci. Paris t. 312 Série I (1991), 725–728
___, Polynômes de Hermite généralisés. C.R. Acad. Sci. Paris t. 313 Série I (1991), 579–582.
Macdonald, I.G., The Volume of a Compact Lie Group. Invent. Math. 56 (1980), 93–95.
___, Some conjectures for root systems. SIAM J. Math. Anal. 13 (1982), 988–1007.
Moser, J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. in Math. 16 (1975), 197–220.
Olshanetsky, M.A., Perelomov, A.M., Completely integrable Hamiltonian systems connected with semisimple Lie algebras. Invent. Math. 37 (1976), 93–108.
___, Quantum systems related to root systems, and radial parts of Laplace operators. Funct. Anal. Appl. 12 (1978), 121–128.
Opdam, E.M., Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group. Compositio Math. 85 (1993), 333–373.
___, Harmonic analysis for certain representations of graded Hecke algebras. Acta Math. 175 (1995), 75–121
Pasquier, V.: A lecture on the Calogero-Sutherland models. In: Integrable models and strings (Espoo, 1993), Lecture Notes in Phys. 436, Springer, 1994, pp. 36–48.
Perelomov, A.M., Algebraical approach to the solution of a one-dimensional model of N interacting particles. Teor. Mat. Fiz. 6 (1971), 364–391.
Polychronakos, A.P., Exchange operator formalism for integrable systems of particles. Phys. Rev. Lett. 69 (1992), 703–705.
Rösler, M., Bessel-type signed hypergroups on R. In: Heyer, H., Mukherjea, A. (eds.) Probability measures on groups and related structures XI. Proceedings, Oberwolfach 1994. World Scientific 1995, 292–304.
___, Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys. 192 (1998), 519–542.
___, Positivity of Dunkl’s intertwining operator. Duke Math. J. 98 (1999), 445–463.
___, Short-time estimates for heat kernels associated with root systems, in Special Functions, Conf. Proc. Hong Kong June 1999, eds. C. Dunkl et al., World Scientific, Singapore, 2000, 309–323.
___, One-parameter semigroups related to abstract quantum models of Calogero type. In: Infinite Dimensional Harmonic Analysis (Kyoto, Sept. 1999, eds. H. Heyer et al.) Gräbner-Verlag 2000, 290–305.
Rösler, M., de Jeu, M., Asymptotic analysis for the Dunkl kernel. math.CA-/0202083; to appear in J. Approx. Theory.
Rösler, M., Voit, M., Markov Processes related with Dunkl operators. Adv. Appl. Math. 21 (1998), 575–643.
Rosenblum, M., Generalized Hermite polynomials and the Bose-like oscillator calculus. In: Operator Theory: Advances and Applications, Vol. 73, Basel, Birkhäuser Verlag 1994, 369–396.
Sutherland, B., Exact results for a quantum many-body problem in one dimension. Phys. Rep. A5 (1972), 1372–1376.
Titchmarsh, E.C., The Theory of Functions. 2nd ed., Oxford University Press, 1950.
Trimèche, K., Paley-Wiener Theorems for the Dunkl transform and Dunkl translation operators. Integral Transform. Spec. Funct. 13 (2002), 17–38.
A. Wintner, On a theorem of Bôcher in the theory of ordinary linear differential equations, Amer. J. Math. 76 (1954), 183–190.
Ujino, H., Wadati., M., Rodrigues formula for Hi-Jack symmetric polynomials associated with the quantum Calogero model. J. Phys. Soc. Japan 65 (1996), 2423–2439.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Rösler, M. (2003). Dunkl Operators: Theory and Applications. In: Koelink, E., Van Assche, W. (eds) Orthogonal Polynomials and Special Functions. Lecture Notes in Mathematics, vol 1817. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44945-0_3
Download citation
DOI: https://doi.org/10.1007/3-540-44945-0_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40375-3
Online ISBN: 978-3-540-44945-4
eBook Packages: Springer Book Archive