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Scattering on the p-adic field and a trace formula

Jean-Francois Burnol
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Abstract

I apply the set-up of Lax-Phillips Scattering Theory to a non-archimedean local field. It is possible to choose the outgoing space and the incoming space to be Fourier transforms of each other. Key elements of the Lax-Phillips theory are seen to make sense and to have the expected interrelations: the scattering matrix S, the projection K to the interacting space, the contraction semi-group Z and the time delay operator T. The scattering matrix is causal, its analytic continuation has the expected poles and zeros, and its phase derivative is the (non-negative) spectral function of T, which is also the restriction to the diagonal of the kernel of K. The contraction semi-group Z is related to S (and T) through a trace formula. Introducing an odd-even grading on the interacting space allows to express the Weil local explicit formula in terms of a ``supertrace''. I also apply my methods to the evaluation of a trace considered by Connes.

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Jean-Francois Burnol. 1999-02-01. Scattering on the p-adic field and a trace formula. https://doi.org/10.1155/s1073792800000040

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