Imaginary Powers of the Dunkl Harmonic Oscillator

TL;DR

This paper investigates the spectral properties of the Dunkl harmonic oscillator associated with a finite reflection group, demonstrating boundedness of its imaginary powers on various Lebesgue spaces.

Contribution

It proves that imaginary powers of the Dunkl harmonic oscillator are bounded on L^p spaces and from L^1 into weak L^1, extending spectral analysis in this context.

Findings

Imaginary powers are bounded on L^p for 1<p<∞

Imaginary powers map L^1 into weak L^1

Advances understanding of spectral properties of Dunkl operators

Abstract

In this paper we continue the study of spectral properties of the Dunkl harmonic oscillator in the context of a finite reflection group on $\mathbb{R}^d$ isomorphic to $\mathbb{Z}^d_2$. We prove that imaginary powers of this operator are bounded on $L^p$, $1<p<\infty$, and from $L^1$ into weak $L^1$.