Explicit Formulas for the One-Parameter Group Generated by the Dunkl Operator on $\mathbb{R}$

Abstract: Let $T_{b}$ be the Dunkl operator for the reflection group $G=\mathbb{Z}/2\mathbb{Z}$, and $D_{b}:=|x|<sup>{b}\,T_{b}\,|x|<sup>{-b}$. We compute explicitly the unitary one-parameter group $e<sup>{tD_{b}}$ generated by $D_{b}$. We obtain two representations: a boundary value representation from the upper and lower half-planes, and a real-variable formula consisting of a translation term and a principal value integral term with an explicit kernel expressed in terms of Legendre functions.

• The paper presents explicit analytic formulas for the one-parameter group e^(tD_b), directly linking Dunkl operator theory with harmonic analysis.
• It develops both boundary-value integral and real-variable formulations using Legendre and Bessel functions to provide precise kernel representations.
• The results imply finite propagation speed and offer practical tools for spectral expansions and applications in quantum integrable systems.

Explicit Formulas for the One-Parameter Group Generated by the Dunkl Operator on $\mathbb{R}$

Introduction

This paper addresses the explicit construction of the unitary one-parameter group $e^{t D_b}$ generated by a conjugate of the Dunkl operator on $L^2(\mathbb{R})$ associated with the $\mathbb{Z}/2\mathbb{Z}$ reflection group. The Dunkl operator and its various generalizations play a significant role in harmonic analysis, special function theory, and intertwined operator analysis due to their rich algebraic and analytic structure. This work provides, for the first time, comprehensive and explicit analytic expressions for the group $e^{t D_b}$, revealing precise kernel representations with strong implications for propagation phenomena, special functions, and spectral theory.

Background and Preliminaries

The Dunkl operator for $G = \mathbb{Z}/2\mathbb{Z}$ is given, up to conjugation, by

$D_b f(x) = \frac{d f}{dx}(x) - \frac{b}{x} f(-x)$

with parameter $b > -\frac{1}{2}$. The natural domain is $|x|^b \mathcal{S}(\mathbb{R})$ where $\mathcal{S}(\mathbb{R})$ is the Schwartz space, ensuring preservation under $e^{t D_b}$0.

The corresponding generalized Fourier transform, the Dunkl transform $e^{t D_b}$1, is also defined via the eigenfunctions of the harmonic oscillator type operator $e^{t D_b}$2 and constructed with normalized Bessel functions. Notably, for $e^{t D_b}$3, $e^{t D_b}$4 reduces to the standard derivative and $e^{t D_b}$5 coincides with the ordinary Fourier transform.

Importantly, $e^{t D_b}$6 satisfies critical intertwining and unitarity properties, notably:

• $e^{t D_b}$7, so $e^{t D_b}$8 is essentially skew-adjoint and generates a unitary group on $e^{t D_b}$9.
• The spectrum and eigenfunctions are described by Laguerre polynomials, analogously to the classical Hermite function scenario.

Main Results: Explicit Representations of $L^2(\mathbb{R})$0

The principal contribution is the explicit formula for $L^2(\mathbb{R})$1. The group is realized both (1) as a boundary-value integral over the complex plane, and (2) as a real-variable formula involving translation and principal value convolution with explicitly constructed kernels, given in terms of Legendre functions. These results can be considered highly detailed functional analytic analogs of the classical Fourier translation formula, with nontrivial structure due to the underlying reflection representation.

Boundary Value Representation

Let $L^2(\mathbb{R})$2 denote a kernel constructed via a combination of Legendre functions of the second kind,

$L^2(\mathbb{R})$3

with

$L^2(\mathbb{R})$4

where $L^2(\mathbb{R})$5 is the Legendre function of the second kind.

Then, the action of $L^2(\mathbb{R})$6 on $L^2(\mathbb{R})$7 is

$L^2(\mathbb{R})$8

providing a full analytic extension to the complex plane and connecting to boundary value distribution theory. For $L^2(\mathbb{R})$9, this reduces to the ordinary translation.

Real-Variable Decomposition

A more "physical space" formula expresses $\mathbb{Z}/2\mathbb{Z}$0 as a translation plus a singular integral (principal value) involving an explicit, $\mathbb{Z}/2\mathbb{Z}$1-dependent, Legendre kernel. For $\mathbb{Z}/2\mathbb{Z}$2,

$\mathbb{Z}/2\mathbb{Z}$3

where

$\mathbb{Z}/2\mathbb{Z}$4

and, more explicitly,

$\mathbb{Z}/2\mathbb{Z}$5

with $\mathbb{Z}/2\mathbb{Z}$6, and $\mathbb{Z}/2\mathbb{Z}$7 are the Legendre functions of the first and second kind, respectively.

Propagation Structure

A key theoretical corollary is that $\mathbb{Z}/2\mathbb{Z}$8 depends only on the values of $\mathbb{Z}/2\mathbb{Z}$9 with $e^{t D_b}$0, which can be seen as an analog of finite speed of propagation, familiar from the classical wave equation, but now in a context involving a nonlocal, reflection-involved operator.

Technical Analysis and Proof Strategy

The derivations employ the generalized spectral theory of $e^{t D_b}$1, an explicit expansion in terms of Laguerre polynomials, and a careful computation of the Dunkl transform's kernel via Bessel and Legendre functions. The analytic machinery involves:

• A decomposition of $e^{t D_b}$2 into positive and negative frequency integrals,
• Application of generalized harmonic analysis, yielding closed Bessel function integrals expressed in terms of Legendre $e^{t D_b}$3,
• Detailed distributional analysis to extract principal value and $e^{t D_b}$4-function contributions arising from analytic continuation and boundary values,
• Nontrivial connection formulas for Gauss hypergeometric functions, essential for evaluating singularities and branch points in the kernels.

These elaborate computations provide highly nontrivial explicit expressions for all cases, including the non-generic regime where $e^{t D_b}$5 or $e^{t D_b}$6 vanish.

Implications and Future Directions

The explicit description of $e^{t D_b}$7 gives full analytic access to spectral and semigroup properties of Dunkl-type operators for reflection groups, which are of foundational interest in harmonic analysis and quantum integrable systems. Immediate applications include fine spectral expansions, precise description of propagation, and kernel estimates for solutions to evolution equations with Dunkl derivatives.

The construction demonstrates that generalized translation and propagation phenomena possess a rich, quantifiable structure even in highly nonlocal and reflection-involved operator settings. The explicit kernels involving special functions may have further relevance for functional inequalities, transmutation operators, and special function theory.

It is anticipated that analogs of these results for higher-dimensional Dunkl operators and more complex reflection groups would exhibit similar phenomena, though with even greater analytic complexity. The methods provided lay groundwork for approaching those more general cases.

Conclusion

The paper provides a complete and explicit analytic solution to the operator evolution problem for the Dunkl operator $e^{t D_b}$8 on $e^{t D_b}$9, characterizing the unitary group $G = \mathbb{Z}/2\mathbb{Z}$0 via boundary-value formulas and real-variable principal value integrals with kernels involving Legendre functions. This advances both the theory and practical computational tools for Dunkl analysis and related areas in mathematics and mathematical physics.