Mathematical Research Letters

Volume 13 (2006)

Number 1

Huygens' principle for the wave equation associated with the trigonometric Dunkl-Cherednik operators

Pages: 43 – 58



Salem Ben Saïd (University of Aarhus)


Let $\a$ be an Euclidean vector space of dimension $N,$ and let $k=(k_\alpha)_{\alpha\in \cal R}$ be a multiplicity function related to a root system $\cal R.$ Let $\Delta(k)$ be the trigonometric Dunkl-Cherednik differential-difference Laplacian. For $(a,t)\in \exp(\a)\times \R,$ denote by $u_k(a,t)$ the solution to the wave equation $\Delta(k) u_k(a,t)=\partial_{tt}u_k(a,t),$ where the initial data are supported inside a ball of radius $R$ about the origin. We prove that $u_k$ has support in the shell $\{(a,t)\in \exp(\a)\times \R\;|\; \vert t\vert-R\leq \Vert \log a\Vert\leq \vert t\vert+R\}$ if and only if the root system $\cal R$ is reduced, $k_\alpha\in \N$ for all $\alpha\in \cal R,$ and $N$ is odd starting from $3.$

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