Abstract:
In Dunkl theory on $\mathbb{R}^{n}$ which generalizes classical Fourier analysis, we study the solution of the Klein–Gordon-equation defined by: $$\partial_{t}^{2}u-\Delta_{k}u=-m^{2}u , \ \ \ u (x,0)=g(x) , \ \ \ \partial_{t}u(x,0)=f(x) $$ with $m > 0$, $\partial_{t}^{2}u$ denoting the second derivative of the solution $u$ with respect to $t$, and $\Delta_{k}u$ the Dunkl Laplacian with respect to $x$ where $f$ and $g$ being two functions in $\mathcal{S}(\mathbb{R}^{n})$ defining the initial conditions. An integral representation for its solution is obtained, which makes it possible to study certain properties. As a specific result, the energies associated with the Dunkl–Klein–Gordon equation are studied.
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