The wave equation for Dunkl operators |
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Authors: | Salem Ben Saïd Bent
rsted |
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Institution: | aUniversité Henri Poincaré-Nancy 1, Institut Elie Cartan, Département de Mathématiques, B.P. 239, 54506 Vandoeuvre-Les-Nancy, Cedex, France;bUniversity of Aarhus, Department of Mathematical Sciences, Building 530, Ny Munkegade, DK 8000, Aarhus C, Denmark |
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Abstract: | Let k = (kα)αε, be a positive-real valued multiplicity function related to a root system , and Δk be the Dunkl-Laplacian operator. For (x, t) ε N, × , denote by uk(x, t) the solution to the deformed wave equation Δkuk,(x, t) = δttuk(x, t), where the initial data belong to the Schwartz space on N. We prove that for k 0 and N l, the wave equation satisfies a weak Huygens' principle, while a strict Huygens' principle holds if and only if (N − 3)/2 + Σαε+kα ε . Here + is a subsystem of positive roots. As a particular case, if the initial data are supported in a closed ball of radius R > 0 about the origin, the strict Huygens principle implies that the support of uk(x, t) is contained in the conical shell {(x, t), ε N × | |t| − R x |t| + R}. Our approach uses the representation theory of the group SL(2, ), and Paley-Wiener theory for the Dunkl transform. Also, we show that the (t-independent) energy functional of uk is, for large |t|, partitioned into equal potential and kinetic parts. |
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