  The scattering theory of Lax and Phillips, designed primarily for hyperbolic
systems, such as electromagnetic or acoustic waves, is described. This theory
provides a realization of the theorem of Foias and Nagy; there is a subspace of
the Hilbert space in which the unitary evolution of the system, restricted to
this subspace, is realized as a semigroup. The embedding of the quantum theory
into this structure, carried out by Flesia and Piron, is reviewed. We show how
the density matrix for an effectively pure state can evolve to an effectively
mixed state (decoherence) in this framework. Necessary conditions are given for
the realization of the relation between the spectrum of the generator of the
semigroup and the singularities of the $S$-matrix (in energy representation).
It is shown that these conditions may be met in the Liouville space formulation
of quantum evolution, and in the Hilbert space of relativistic quantum theory.
