Imaginary Powers of the Dunkl Harmonic Oscillator

Adam Nowak, Krzysztof Stempak

发表信息：: SIGMA 5 (2009), 016, 12 pages
DOI：: 10.3842/SIGMA.2009.016

英文摘要：

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 $L^1$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Riesz transforms for the Dunkl harmonic oscillator

We define and investigate a system of Riesz transforms related to the Dunkl harmonic oscillator.

2008-02-04math.CA

Free boson realization of the Dunkl intertwining operator in one dimension

The operator that intertwines between the $\mathbb{Z}_2$ - Dunkl operator and the derivative is shown to have a realization in terms of the oscillator operators in one dimension. This observation rests on the fact that the Dunkl intertwining operator maps the Hermite polynomials on the generalized Hermite polynomials.

2022-01-06math-phmath.CAmath.MP

On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to $\mathbb{Z}_2^d$. Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators, $g$-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calderón-Zygmund theory we prove that these operators are bounded on weighted $L^p$ spaces, $1 < p < \infty$, and from weighted $L^1$ to weighted weak $L^1$. …

2016-09-29math.CA

Imaginary powers of $(k,1)$-generalized harmonic oscillator

In this paper we will define and investigate the imaginary powers $\left(-\triangle_{k,1}\right)^{-iσ},σ\in\mathbb{R}$ of the $(k,1)$-generalized harmonic oscillator $-\triangle_{k,1}=-\left\|x\right\|\triangle_k+\left\|x\right\|$ and prove the $L^p$-boundedness $(1<p<\infty)$ and weak $L^1$-boundedness of such operators. It is a parallel result to the $L^p$-boundedness $(1<p<\infty)$ and weak $L^1$-boundedness of the imaginary powers of the Dunkl harmonic oscillator $-\triangle_k+\left\|x\right\|^2$. To prove this result, we develop the Calderón--Zygmund theory adapted to the $(k,1)$-generali…

2022-07-08math.CA

Hardy-type inequalities for fractional powers of the Dunkl--Hermite operator

We prove Hardy-type inequalities for a fractional Dunkl--Hermite operator which incidentally give Hardy inequalities for the fractional harmonic oscillator as well. The idea is to use $h$-harmonic expansions to reduce the problem in the Dunkl--Hermite context to the Laguerre setting. Then, we push forward a technique based on a non-local ground representation, initially developed by R. L. Frank, E. H. Lieb and R. Seiringer in the Euclidean setting, to get a Hardy inequality for the fractional-type Laguerre operator. The above-mentioned method is shown to be adaptable to an abstract setting, wh…

2016-09-05math.CAmath.FA

Hardy inequalities for fractional $(k,a)$-generalized harmonic oscillator

In this paper, we will define $a$-deformed Laguerre operators $L_{a,α}$ and $a$-deformed Laguerre holomorphic semigroups on $L^2\left(\left(0,\infty\right),dμ_{a,α}\right)$. Then we give a spherical harmonic expansion, which reduces to the Bochner-type identity when taking the boundary value $z=\frac{πi}2$, of the $(k,a)$-generalized Laguerre semigroup introduced by S. Ben Saïd, T. Kobayashi and B. Ørsted. And then we prove a Hardy inequality for fractional powers of the $a$-deformed Dunkl harmonic oscillator $\triangle_{k,a}:=\left|x\right|^{2-a}\triangle_k-\left|x\right|^a$ using this expans…

2022-10-27math.CAmath.RT

更多相关论文收起更多论文

具有二十面体对称性的量子谐振子及若干显式波函数

The quantum harmonic oscillator with icosahedral symmetry and some explicit wavefunctions

本文研究H3型正二十面体对称群下的修正量子谐振子，利用Dunkl拉普拉斯算子定义哈密顿量，构造两类特殊波函数并显式计算其平方范数，得到对称不变多项式，还发现了与哈密顿量对易的六阶算子。

2023-02-21math-phmath.CAmath.MP

Mass-sign duality of cubic oscillators

Two imaginary cubic anharmonic oscillators with the respective positive and negative mass term are shown isospectral.

2006-12-22math-phmath.CAmath.MP