Lax-Phillips evolutions in quantum mechanics and two-space scattering

Canonical extensions of “physical Hamiltonians” are introduced. They are generators of Lax-Phillips (LP-)evolutions. Restricted to the outgoing/incoming subspace an LP-evolution is a contractive semigroup. Spectral properties of the generator and its adjoint are mentioned. Characteristic conditions for the identification operator of a two-space scattering system, consisting of a unitary strongly continuous evolution U(R) on a Hilbert space H and the so-called reference LP-evolution on L2(R, dx, K), are presented in such a way that {U(R), H} is an LP-evolution.