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Scattering on the p-adic field and a trace formula
I apply the set-up of Lax-Phillips Scattering Theory to a nonarchimedean 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 semigroup 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 (nonnegative) spectral function of T, which is also the restriction to the diagonal of the kernel of K. The contraction semigroup Z is related to S (and T) through a trace formula. Introducing an odd-even grading on the interacting space allows the expression of the Weil local explicit formula in terms of a "supertrace." I also evaluate a trace first considered by Connes.