  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.
