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

Temma Aoyama

Abstract

Let $T_{b}$ be the Dunkl operator for the reflection group $G=\mathbb{Z}/2\mathbb{Z}$, and $D_{b}:=|x|^{b}\,T_{b}\,|x|^{-b}$. We compute explicitly the unitary one-parameter group $e^{tD_{b}}$ 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.

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

Abstract

Let be the Dunkl operator for the reflection group , and . We compute explicitly the unitary one-parameter group generated by . 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.

Paper Structure

This paper contains 13 sections, 12 theorems, 77 equations.

Table of Contents

  1. Statement of the Main Result
  2. Review of Background Material
  3. Main Theorem
  4. Proof of the Main Theorem
  5. Outline of the Proof
  6. Positive and Negative Frequency Decomposition
  7. An Integral Formula Involving Bessel Functions
  8. Boundary Value Representation
  9. Computation of the Contribution from the $\delta$-Term
  10. Real-Variable Formula
  11. Evaluation of $K_{b}(x,y;t)$
  12. Appendix
  13. A Connection Formula for ${}_2F_{1}(\alpha,\beta;\gamma;z)$ in the Case $\gamma=\alpha+\beta$

Key Result

Proposition 2.3.2

For $b >-\frac{1}{2}$ and $\mathrm{Im}(z) <0$, $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (18)

  • Remark 1.2.1: An analogue of the finite propagation property
  • Remark 1.2.2: Expanded form of Main Theorem B
  • Remark 2.3.1: Properties of $\Psi_{b}\left(w\right)$
  • Proposition 2.3.2
  • Lemma 2.3.4
  • Lemma 2.3.5
  • Theorem 2.4.1: Boundary Value Representation
  • Remark 2.4.2: The case when $b=0$
  • Remark 2.4.3: Behavior at $x=0$
  • Proposition 2.5.1
  • ...and 8 more