## DUNKL OPERATORS: THEORY AND APPLICATIONS (2002)

Citations: | 20 - 1 self |

### BibTeX

@MISC{Rösler02dunkloperators:,

author = {Margit Rösler},

title = {DUNKL OPERATORS: THEORY AND APPLICATIONS},

year = {2002}

}

### OpenURL

### 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.

### Citations

524 |
An Introduction to Orthogonal Polynomials
- Chihara
- 1978
(Show Context)
Citation Context ... with respect to [ . , . ]k . The associated Hermite polynomials are given, up to multiplicative constants, by the generalized Hermite polynomials Hk n(x/ √ 2) on R. These polynomials can be found in =-=[7]-=- and were further studied in [56] in connection with a Bose-like oscillator calculus. The Hk n are orthogonal with respect to |x| 2ke−|x|2 and can be written as � Hk 2n(x) = (−1) n22nn! L k−1/2 n (x2 ... |

459 |
Reflection Groups and Coxeter Groups
- Humphreys
- 1990
(Show Context)
Citation Context ...all call Dunkl operators for short, and to the Dunkl transform. General references are [11]-[15], [18], [30] and [44]; for a background on reflection groups and root systems the reader is referred to =-=[29]-=- and [23]. We do not intend to give a complete survey, but rather focus on those aspects which will be important in the context of this lecture series. 2.1 Root systems and reflection groups The basic... |

342 |
Groups and Geometric Analysis
- Helgason
- 1984
(Show Context)
Citation Context ...σj . + k0 · � � 1 − σij + 1 − τij � j�=i xi − xj xi + xj (i = 1, . . . , N), 2.3 A formula of Macdonald and its analog in Dunkl theory In the classical theory of spherical harmonics (see for instance =-=[28]-=-) the following bilinear pairing on Π , sometimes called Fischer product, plays an important role: [p, q]0 := (p(∂)q)(0), p, q ∈ Π. This pairing is closely related to the scalar product in L 2 (R N , ... |

337 |
Foundations of Modern Probability
- Kallenberg
- 2002
(Show Context)
Citation Context ...l of the tentative generator. The tool is the following useful variant of the Lumer-Phillips theorem, which characterizes Feller-Markov semigroups in terms of a “positive maximum principle”, see e.g. =-=[34]-=-, Theorem 17.11. In fact, this Theorem motivated the positive minimum principle of Theorem 4.3 in the positivity-proof for Vk .sDunkl Operators: Theory and Applications 33 Theorem 4.7. Let (A, D(A)) b... |

195 |
The theory of functions
- Titchmarsh
- 1939
(Show Context)
Citation Context .... In the particular case g = e (the unit of G), this latter result can be extended to a larger range of complex arguments by use of the PhragménLindelöf principle for the right half plane H (see e.g. =-=[58]-=-): Proposition 5.1. Suppose f : H → C is analytic and regular in H ∩ {z ∈ C : |z| > R} for some R > 0 with limt→∞ f(it) = a, limt→∞ f(−it) = b and for each δ > 0, f(z) = O(e δ|z| ) as z → ∞ within H. ... |

124 |
Three Integrable Hamiltonian Systems Connected with Isospectral Deformations
- Moser
- 1975
(Show Context)
Citation Context ...ebraically independent symmetric linear operators in L 2 (R N ) including HC . We mention that the complete integrability of the classical Hamiltonian systems associated with (3.1) goes back to Moser =-=[41]-=-. There exist generalizations of the classical Calogero-Moser-Sutherland models in the context of abstract root systems, see for instance [42], [43]. In particular, if R is an arbitrary root system on... |

117 |
Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials
- Calogero
- 1971
(Show Context)
Citation Context ...mal field theory, and they are being used to test the ideas of fractional statistics ([24], [25]). While explicit spectral resolutions of such models were already obtained by Calogero and Sutherland (=-=[6]-=-, [57]), a new aspect in the understanding of their algebraic structure and quantum integrability was much later initiated by [48] and [26]. The Hamiltonian under consideration is hereby modified by c... |

117 | Differential-difference operators associated to reflection groups - Dunkl - 1989 |

97 |
Exact results for a quantum many-body problem in one dimension
- Sutherland
- 1972
(Show Context)
Citation Context ...ield theory, and they are being used to test the ideas of fractional statistics ([24], [25]). While explicit spectral resolutions of such models were already obtained by Calogero and Sutherland ([6], =-=[57]-=-), a new aspect in the understanding of their algebraic structure and quantum integrability was much later initiated by [48] and [26]. The Hamiltonian under consideration is hereby modified by certain... |

88 |
Spectral theory and differential operators
- Davies
- 1995
(Show Context)
Citation Context ...genvalues λn ∈ R, then it is straightforward that A is essentially selfadjoint, and that the spectrum of the self-adjoint operator A is given by σ(A) = {λn , n ∈ Z+}. (See for instance Lemma 1.2.2 of =-=[8]-=-).s22 Margit Rösler In our situation, the operator Hk is densely defined and symmetric in L 2 (R N , wk) (the first order Dunkl operators being anti-symmetric), and the same holds for Jk in L 2 (R N ,... |

80 | A recursion and a combinatorial formula for Jack polynomials
- Knop, Sahi
(Show Context)
Citation Context ...system {ϕν}, made up by the so-called non-symmetric Jack polynomials. For a multiplicity parameter k > 0, the associated non-symmetric Jack polynomials Eν , ν ∈ ZN + , as introduced in [45] (see also =-=[35]-=-), are uniquely defined by the following conditions: (i) Eν(x) = x ν + � cν, µx µ with cν,µ ∈ R; µ<P νs24 Margit Rösler (ii) For all µ <P ν , (Eν(x), x µ ) k = 0 . Here <P is a dominance order defined... |

72 | Some conjectures for root systems - Macdonald - 1982 |

70 |
transforms associated to finite reflection groups, Proc. of the special session on hypergeometric functions on domains of positivity, Jack polynomials and applications
- Dunkl, Hankel
- 1992
(Show Context)
Citation Context ...k ≥ 0. Then for p ∈ Π , y, z ∈ CN , � (1) RN � (2) e −∆k/2 p (x) Ek(x, y) e −|x|2 /2 wk(x)dx = ck e 〈y,y〉/2 p(y), RN Ek(x, y) Ek(x, z) e −|x|2 /2 wk(x)dx = ck e (〈y,y〉+〈z,z〉)/2 Ek(y, z). Proof. (c.f. =-=[15]-=-.) We shall use the Macdonald-type formula (2.5) for the pairing [ . , . ]k . First, we prove that In fact, if p ∈ Pn, then [E (n) k (x, . ), . ]k = p(x) for all p ∈ Pn, x ∈ R N . (2.11) p(x) = (〈x, ∂... |

69 | The Calogero-Sutherland model and generalized classical polynomials
- BAKER, FORRESTER
- 1997
(Show Context)
Citation Context ... square type. They are exactly solvable and have gained considerable interest in theoretical physics during the last years. Among the broad literature in this area, we refer to [10], [36], [33], [5], =-=[2]-=--[4], [46], [47], [61], [17]. CMS models have in particular attracted some attention in conformal field theory, and they are being used to test the ideas of fractional statistics ([24], [25]). While e... |

60 | Integral kernels with reflection group invariance - Dunkl - 1991 |

54 |
Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group
- Opdam
- 1993
(Show Context)
Citation Context ...on is to provide an introduction to the theory of rational Dunkl operators, which we shall call Dunkl operators for short, and to the Dunkl transform. General references are [11]-[15], [18], [30] and =-=[44]-=-; for a background on reflection groups and root systems the reader is referred to [29] and [23]. We do not intend to give a complete survey, but rather focus on those aspects which will be important ... |

51 |
Completely integrable Hamiltonian systems connected with semisimple Lie algebras
- Olshanetsky, Perelomov
- 1976
(Show Context)
Citation Context ...iltonian systems associated with (3.1) goes back to Moser [41]. There exist generalizations of the classical Calogero-Moser-Sutherland models in the context of abstract root systems, see for instance =-=[42]-=-, [43]. In particular, if R is an arbitrary root system on R N and k is a nonnegative multiplicity function on it, then the corresponding abstract Calogero Hamiltonian with harmonic confinement is giv... |

49 |
Exact operator solution of the CalogeroSutherland
- LAPOINTE, VINET
- 1996
(Show Context)
Citation Context ...ntials of inverse square type. They are exactly solvable and have gained considerable interest in theoretical physics during the last years. Among the broad literature in this area, we refer to [10], =-=[36]-=-, [33], [5], [2]-[4], [46], [47], [61], [17]. CMS models have in particular attracted some attention in conformal field theory, and they are being used to test the ideas of fractional statistics ([24]... |

47 |
The Asymptotic Solution of Linear Differential Systems: Applications of the Levinson Theorem
- Eastham
- 1989
(Show Context)
Citation Context ...atisfies the conditions of the following classical theorem on the asymptotic integration of ordinary differential equations (hereby of course, the admissibility conditions on κ come in). Theorem 5.4. =-=[20]-=-, [60]. Consider the linear differential equation x ′ (t) = A(t)x(t), (5.3) where A : [t0, ∞) → Cn×n is a continuous matrix-valued function satisfying the following integrability conditions: (1) The m... |

43 |
The Dunkl transform
- Jeu
- 1993
(Show Context)
Citation Context ...his section is to provide an introduction to the theory of rational Dunkl operators, which we shall call Dunkl operators for short, and to the Dunkl transform. General references are [11]-[15], [18], =-=[30]-=- and [44]; for a background on reflection groups and root systems the reader is referred to [29] and [23]. We do not intend to give a complete survey, but rather focus on those aspects which will be i... |

43 | Positivity of Dunkl’s intertwining operator - RÖSLER - 1999 |

39 | Generalized Hermite polynomials and the heat equation for Dunkl operators - Rösler - 1998 |

39 | The Theory of Functions, 2nd Ed - Titchmarsh - 1939 |

34 |
A remark on the Dunkl differential-difference operators, Harmonic analysis on reductive groups
- Heckman
- 1991
(Show Context)
Citation Context ... ∈ Π : g · p = p for all g ∈ G}. If p ∈ Π G , then it follows from the equivariance of the Dunkl operators (Exercise 3) that p(T ) commutes with the G-action; a detailed argument for this is given in =-=[26]-=-. Thus p(T ) leaves Π G invariant, and we obtain in particular that for fixed y ∈ C N , the generalized Bessel function Jk( . , y) is a solution to the following Bessel-system: p(T )f = p(y)f for all ... |

32 |
Exchange operator formalism for integrable systems of particles, Phys
- Polychronakos
(Show Context)
Citation Context ...tions of such models were already obtained by Calogero and Sutherland ([6], [57]), a new aspect in the understanding of their algebraic structure and quantum integrability was much later initiated by =-=[48]-=- and [26]. The Hamiltonian under consideration is hereby modified by certain exchange operators, which allow to write it in a decoupled form. These exchange modifications can be expressed in terms of ... |

30 |
Generalized Hermite polynomials and the Bose-like oscillator calculus, in Nonselfadjoint Operators and Related Topics (Beer Sheva
- Rosenblum
- 1992
(Show Context)
Citation Context ...e associated Hermite polynomials are given, up to multiplicative constants, by the generalized Hermite polynomials Hk n(x/ √ 2) on R. These polynomials can be found in [7] and were further studied in =-=[56]-=- in connection with a Bose-like oscillator calculus. The Hk n are orthogonal with respect to |x| 2ke−|x|2 and can be written as � Hk 2n(x) = (−1) n22nn! L k−1/2 n (x2 ), i=1 Hk 2n+1(x) = (−1) n22n+1n!... |

28 |
Singular polynomials for finite reflection groups
- Dunkl, Jeu, et al.
- 1994
(Show Context)
Citation Context ...m of this section is to provide an introduction to the theory of rational Dunkl operators, which we shall call Dunkl operators for short, and to the Dunkl transform. General references are [11]-[15], =-=[18]-=-, [30] and [44]; for a background on reflection groups and root systems the reader is referred to [29] and [23]. We do not intend to give a complete survey, but rather focus on those aspects which wil... |

27 | Markov processes related with Dunkl operator
- RÖSLER, VOIT
- 1998
(Show Context)
Citation Context ...nerates a strongly continuous and positivitypreserving contraction semigroup on L2 (RN , wk) which is given by � H(t)f(x) = Γk(t, x, y)f(y)wk(y)dy , (t > 0). R N Theorem 4.8 was the starting point in =-=[55]-=- to construct an associated FellerMarkov process on R N which can be considered a generalization of the usual Brownian motion. The transition probabilities of this process are defined in terms of a se... |

26 |
Reflection groups and orthogonal polynomials on the sphere,Math.Z.197(1988
- Dunkl
(Show Context)
Citation Context ...a finite reflection group on a Euclidean space. The first class of such operators, now often called “rational” Dunkl operators, were introduced by C.F. Dunkl in the late 80ies. In a series of papers (=-=[11]-=--[15]), he built up the framework for a theory of special functions and integral transforms in several variables related with reflection groups. Since then, various other classes of Dunkl operators ha... |

25 |
Paley-Wiener Theorems for the Dunkl transform and Dunkl translation operators
- TRIMÈCHE
(Show Context)
Citation Context ...ws (c.f. [50]): τyf(x) := 1 � �f k (ξ) Ek(ix, ξ)Ek(iy, ξ) wk(ξ)dξ; y ∈ R N . (4.11) ck R N In the same way, this could be done in L 2 (R N , wk). A powerful extension to C ∞ (R N ) is due to Trimèche =-=[59]-=-. Note that in case k = 0, we simplysDunkl Operators: Theory and Applications 31 have τyf(x) = f(x + y). In the rank-one case, the above translation coincides with the convolution on a so-called signe... |

23 | Non–symmetric Jack polynomials and integral kernels. q-alg/9612003 - Baker, Forrester |

20 |
Polynômes de Laguerre généralisés
- Lassalle
- 1991
(Show Context)
Citation Context ...als Eν are also orthogonal with respect to the Dunkl pairing [ . , . ]k ; for details see [50]. The corresponding generalized Hermite polynomials and their symmetric counterparts have been studied in =-=[37]-=-, [38] and in [2] - [4]. As an immediate consequence of Theorem 3.1 we obtain analogues of the classical second order differential equations for generalized Hermite polynomials and Hermite functions: ... |

19 | The Calogero model - anyonic representation, fermionic extension and supersymmetry
- Brink, Hansson, et al.
- 1993
(Show Context)
Citation Context ...verse square type. They are exactly solvable and have gained considerable interest in theoretical physics during the last years. Among the broad literature in this area, we refer to [10], [36], [33], =-=[5]-=-, [2]-[4], [46], [47], [61], [17]. CMS models have in particular attracted some attention in conformal field theory, and they are being used to test the ideas of fractional statistics ([24], [25]). Wh... |

18 | Polynômes de Hermite généralisés - Lassalle - 1991 |

17 |
Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement
- Diejen
- 1997
(Show Context)
Citation Context ...extensions to our general setting. We conclude this section with a list of them. The proofs can be found in [50]. For further results on generalized Hermite polynomials, one can also see for instance =-=[9]-=-. Theorem 3.2. Let {Hν} be the Hermite polynomials associated with the basis {ϕν} on R N and let x, y ∈ R N . Then (1) (Rodrigues formula) Hν(x) = (−1) |ν| e |x|2 /2 ϕν(T )e −|x|2 /2 (2) (Generating r... |

17 | Common algebraic structure for the Calogero–Sutherland models
- Kakei
- 1996
(Show Context)
Citation Context ... of inverse square type. They are exactly solvable and have gained considerable interest in theoretical physics during the last years. Among the broad literature in this area, we refer to [10], [36], =-=[33]-=-, [5], [2]-[4], [46], [47], [61], [17]. CMS models have in particular attracted some attention in conformal field theory, and they are being used to test the ideas of fractional statistics ([24], [25]... |

17 |
Algebraic approach to the solution of a one-dimensional model of N interacting particles
- Perelomov
- 1971
(Show Context)
Citation Context ...hey are exactly solvable and have gained considerable interest in theoretical physics during the last years. Among the broad literature in this area, we refer to [10], [36], [33], [5], [2]-[4], [46], =-=[47]-=-, [61], [17]. CMS models have in particular attracted some attention in conformal field theory, and they are being used to test the ideas of fractional statistics ([24], [25]). While explicit spectral... |

16 | The Calogero-Sutherland model and polynomials with prescribed symmetry - Baker, Forrester - 1997 |

15 |
Bessel-type signed hypergroups on R
- Rösler
- 1994
(Show Context)
Citation Context ...perators: Theory and Applications 31 have τyf(x) = f(x + y). In the rank-one case, the above translation coincides with the convolution on a so-called signed hypergroup structure which was defined in =-=[49]-=-; see also [56]. Similar structures are not yet known in higher rank cases. Clearly, τyf(x) = τxf(y); moreover, the inversion theorem for the Dunkl transform assures that τ0f = f and (τyf) ∧k (ξ) = Ek... |

14 | Intertwining operators associated to the group S3 - Dunkl - 1995 |

12 | Quantum systems related to root systems, and radial parts of Laplace operators - Olshanetsky, Perelomov - 1978 |

11 |
Calogero-Sutherland-Moser Models
- Diejen, Vinet
- 2000
(Show Context)
Citation Context ...ese describe algebraically integrable systems in one dimension and have gained considerable interest in mathematical physics, especially in conformal field theory. A good bibliography is contained in =-=[10]-=-. The aim of these lecture notes is an introduction to rational Dunkl theory, with an emphasis on the author’s results in this area. Rational Dunkl operators bear a rich analytic structure which is no... |

8 |
Exact dynamical correlation functions of the Calogero-Sutherland model and one dimensional fractional statistics in one dimension: View from an exactly solvable model
- Ha
- 1995
(Show Context)
Citation Context ...[36], [33], [5], [2]-[4], [46], [47], [61], [17]. CMS models have in particular attracted some attention in conformal field theory, and they are being used to test the ideas of fractional statistics (=-=[24]-=-, [25]). While explicit spectral resolutions of such models were already obtained by Calogero and Sutherland ([6], [57]), a new aspect in the understanding of their algebraic structure and quantum int... |

6 |
Orthogonal Polynomials of Several Variables,” Cambridge
- Dunkl, Xu
- 2001
(Show Context)
Citation Context ...lso g(α) = α for all α ∈ R, then g must be the identity. This implies assertion (1) because the order of S(R) is finite. Property (2) is more involved. An elegant proof can be found in Section 4.2 of =-=[19]-=-. Exercise 1. If g ∈ O(N, R) and α ∈ R N \ {0}, then g σαg −1 = σgα. Together with part (2) of the previous lemma, this shows that there is a bijective correspondence between the conjugacy classes of ... |

6 | Operators commuting with Coxeter group actions on polynomials - Dunkl - 1990 |

5 |
Rodorigues formula for Hi-Jack symmetric polynomials associated with the quantum Calogero
- Ujino, Wadati
- 1996
(Show Context)
Citation Context ...e exactly solvable and have gained considerable interest in theoretical physics during the last years. Among the broad literature in this area, we refer to [10], [36], [33], [5], [2]-[4], [46], [47], =-=[61]-=-, [17]. CMS models have in particular attracted some attention in conformal field theory, and they are being used to test the ideas of fractional statistics ([24], [25]). While explicit spectral resol... |

4 |
Finite Reflection Groups; Second edition
- Grove, Benson
- 1985
(Show Context)
Citation Context ...Dunkl operators for short, and to the Dunkl transform. General references are [11]-[15], [18], [30] and [44]; for a background on reflection groups and root systems the reader is referred to [29] and =-=[23]-=-. We do not intend to give a complete survey, but rather focus on those aspects which will be important in the context of this lecture series. 2.1 Root systems and reflection groups The basic ingredie... |

4 |
The Volume of a Compact Lie
- Macdonald
- 1980
(Show Context)
Citation Context ...etimes called Fischer product, plays an important role: [p, q]0 := (p(∂)q)(0), p, q ∈ Π. This pairing is closely related to the scalar product in L 2 (R N , e −|x|2 /2 dx); in fact, in his short note =-=[39]-=- Macdonald observed the following identity: [p, q]0 = (2π) −N/2 � e −∆/2 p(x) e −∆/2 q(x) e −|x|2 /2 dx. R N Here e−∆/2 is well-defined as a linear operator on Π by means of the terminating series e −... |

4 | de Jeu, Asymptotic analysis for the Dunkl kernel
- Rösler, M
(Show Context)
Citation Context ...when one of its arguments is fixed and the other tends to infinity either within a Weyl chamber of the associated reflection group, or within a suitable complex domain. These results are contained in =-=[54]-=-. They generalize the well-known asymptotics of the confluent hypergeometric function 1F1 to the higher-rank setting. One motivation to study the asymptotics of Ek is to determine the asymptotic behav... |

4 | Dunkl operators, Séminaire Bourbaki 828, 1996–97; Astérisque 245 - Heckman - 1997 |