Spacetime causality in the study of the Hankel transform
نویسنده
چکیده
We study Hilbert space aspects of the Klein-Gordon equation in two-dimensional spacetime. We associate to its restriction to a spacelike wedge a scattering from the past light cone to the future light cone, which is then shown to be (essentially) the Hankel transform of order zero. We apply this to give a novel proof, solely based on the causality of this spatio-temporal wave propagation, of the theorem of de Branges and V. Rovnyak concerning Hankel pairs with a support property. We recover their isometric expansion as an application of Riemann’s general method for solving CauchyGoursat problems of hyperbolic type. keywords: Klein-Gordon equation; Hankel and Fourier transforms; Scattering. Université Lille 1 UFR de Mathématiques Cité scientifique M2 F-59655 Villeneuve d’Ascq France [email protected] v1: 25 Sept. 2005; Final: 5 March 2006
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