Spacetime causality in the study of the Hankel transform

Details Spacetime causality in the study of the Hankel transform Spacetime causality in the study of the Hankel transform

2005-09-27

arxiv.org/abs/math/0509619v4

Ann. Henri Poincar\'e 7 (2006), 1013--1034

Mathematics

Number Theory

24 pages. Final ms, to appear. Improvements on pages 8 and 9, and 15-16

Scientific paper

10.1007/s00023-006-0274-8

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 Cauchy-Goursat problems of hyperbolic type.

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