  This paper treats the dynamics and scattering of a model of coupled
oscillating systems, a finite dimensional one and a wave field on the half
line. The coupling is realized producing the family of selfadjoint extensions
of the suitably restricted self-adjoint operator describing the uncoupled
dynamics. The spectral theory of the family is studied and the associated
quadratic forms constructed. The dynamics turns out to be Hamiltonian and the
Hamiltonian is described, including the case in which the finite dimensional
systems comprises nonlinear oscillators; in this case the dynamics is shown to
exist as well. In the linear case the system is equivalent, on a dense
subspace, to a wave equation on the half line with higher order boundary
conditions, described by a differential polynomial $p(\partial_x)$ explicitely
related to the model parameters. In terms of such structure the Lax-Phillips
scattering of the system is studied. In particular we determine the incoming
and outgoing translation representations, the scattering operator, which turns
out to be unitarily equivalent to the multiplication operator given by the
rational function $-p(i\kappa)^*/p(i\kappa)$, and the Lax-Phillips semigroup,
which describes the evolution of the states which are neither incoming in the
past nor outgoing in the future.
