18 research outputs found
Asymptotic analysis for the Dunkl kernel
This paper studies the asymptotic behavior of the integral kernel of the
Dunkl transform, the so-called Dunkl kernel, 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. The obtained results are
based on the asymptotic analysis of an associated system of ordinary
differential equations. They generalize the well-known asymptotics of the
confluent hypergeometric function to the higher-dimensional
setting and include a complete short-time asymptotics for the Dunkl-type heat
kernel. As an application, it is shown that the representing measures of
Dunkl's intertwining operator are generically continuous.Comment: LaTeX2e, 16 pages, 1 figure. Second and final version, with minor
corrections. Mathematically identical to first version. Accepted by Journal
of Approximation Theor
Support properties of the intertwining and the mean value operators in Dunkl's analysis
In this paper we show that the Dunkl intertwining operator has a compact support which is invariant by the associated Coxeter-Weyl group. This property enables us to determine explicitely the support of the volume mean value operator, a fundamental tool for the study of harmonic functions relative to the Dunkl-Laplacian operator
Three results in Dunkl theory
In this article, we establish first a geometric Paley-Wiener theorem for the
Dunkl transform in the crystallographic case. Next we obtain an optimal bound
for the norm of Dunkl translations in dimension 1. Finally we
describe more precisely the support of the distribution associated to Dunkl
translations in higher dimension
Paley-Wiener theorems for the Dunkl transform
We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl
transform and prove three instances thereof, one of which involves a limit
transition from Opdam's results for the graded Hecke algebra. Furthermore, the
connection between Dunkl operators and the Cartan motion group is established.
It is shown how the algebra of radial parts of invariant differential operators
can be described explicitly in terms of Dunkl operators, which implies that the
generalized Bessel functions coincide with the spherical functions. In this
context, the restriction of Dunkl's intertwining operator to the invariants can
be interpreted in terms of the Abel transform. Using shift operators we also
show that, for certain values of the multiplicities of the restricted roots,
the Abel transform is essentially inverted by a differential operator.Comment: LaTeX, 26 pages, no figures. References updated and minor changes,
mathematically identical to the first version. To appear in Trans. Amer.
Math. So
The Dunkl kernel and intertwining operator for dihedral groups
Dunkl operators associated with finite reflection groups generate a
commutative algebra of differential-difference operators. There exists a unique
linear operator called intertwining operator which intertwines between this
algebra and the algebra of standard differential operators. There also exists a
generalization of the Fourier transform in this context called Dunkl transform.
In this paper, we determine an integral expression for the Dunkl kernel, which
is the integral kernel of the Dunkl transform, for all dihedral groups. We also
determine an integral expression for the intertwining operator in the case of
dihedral groups, based on observations valid for all reflection groups. As a
special case, we recover the result of [Xu, Intertwining operators associated
to dihedral groups. Constr. Approx. 2019]. Crucial in our approach is a
systematic use of the link between both integral kernels and the simplex in a
suitable high dimensional space.Comment: Revision version. A missing factor in formula (6) for the Bochner
expression is added. Related formulas and some typos are corrected. All
comments are welcom
On the Path Integral Formulation of Wigner-Dunkl Quantum Mechanics
Feynman's path integral approach is studied in the framework of the
Wigner-Dunkl deformation of quantum mechanics. We start with reviewing some
basics from Dunkl theory and investigate the time evolution of a Gaussian wave
packet, which exhibits the same dispersion relation as observed in standard
quantum mechanics. Feynman's path integral approach is then extended to
Wigner-Dunkl quantum mechanics. The harmonic oscillator problem is solved
explicitly. We then look at the Euclidean time evolution and the related Dunkl
process. This process, which exhibit jumps, can be represented by two
continuous Bessel processes, one with reflection and one with absorbtion at the
origin. The Feynman-Kac path integral for the harmonic oscillator problem is
explicitly calculated.Comment: Typos corrected and DOI added to reference