Topic
Dunkl operator
About: Dunkl operator is a research topic. Over the lifetime, 393 publications have been published within this topic receiving 7434 citations.
Papers published on a yearly basis
Papers
TL;DR: In this paper, an orthogonal polynomials of Gegenbauer and Jacobi type are studied in the context of root systems and Coxeter groups, and a formula and a bound for the Poisson kernel are obtained for the poisson kernel.
Abstract: Root systems and Coxeter groups are important tools in multivariable analysis. This paper is concerned with differential-difference and integral operators, and orthogonality structures for polynomials associated to Coxeter groups. For each such group, the structures allow as many parameters as the number of conjugacy classes of reflections. The classical orthogonal polynomials of Gegenbauer and Jacobi type appear in this theory as two-dimensional cases. For each Coxeter group and admissible choice of parameters there is a structure analogous to spherical harmonics which relies on the connection between a Laplacian operator and orthogonality on the unit sphere with respect to a group-invariant measure. The theory has been developed in several papers of the author [4,5,6,7]. In this paper, the emphasis is on the study of an intertwining operator which allows the transfer of certain results about ordinary harmonic polynomials to those associated to Coxeter groups. In particular, a formula and a bound are obtained for the Poisson kernel.
545 citations
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TL;DR: In this article, the eigenfunctions of Dunkl's operators were obtained and the spectral problem for these operators was solved for the first time, and the authors obtained an estimate for the spectral properties of these operators.
Abstract: We obtain estimates for the eigenfunctions of Dunkl's operators and solve the spectral problem for these operators.
521 citations
TL;DR: In this paper, the authors introduce the theory of rational Dunkl operators and associated special functions, with an emphasis on positivity and asymptotics, and discuss integrable particle systems of the Calogero-Moser-Sutherland type.
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
394 citations
TL;DR: In this article, a formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived.
Abstract: The wave functions of the Calogero-Sutherland model are known to be expressible in terms of Jack polynomials. A formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived. The creation operators that enter in this formula of Rodrigues-type for the Jack polynomials involve Dunkl operators.
317 citations
TL;DR: Based on the theory of Dunkl operators, the authors presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on ℝ N fixme.
Abstract: Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on ℝ
N
. The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In the case of the symmetric group S
N
, our setting includes the polynomial eigenfunctions of certain Calogero-Sutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive one-parameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem
involves again the kernel K, which is, on the way, proven to be nonnegative for real arguments.
290 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 2 |
| 2021 | 2 |
| 2020 | 12 |
| 2019 | 7 |
| 2018 | 8 |
| 2017 | 26 |