Scattering on the p-adic field and a trace formula

Details Scattering on the p-adic field and a trace formula Scattering on the p-adic field and a trace formula

1999-01-12

arxiv.org/abs/math/9901051v2

Internat. Math. Res. Notices, 2000 No.2, pp57-70

Mathematics

Number Theory

17 pages, plain TeX. v2 adds the evaluation of a trace considered by Connes

Scientific paper

10.1155/S1073792800000040

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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