Let U0(t) and U(t)
be the one-parameter groups governing the time development of solutions
of the Klein-Gordon equation, □φ = m2φ, and the perturbed equation,
□φ = m2φ + V (x)φ, respectively. In a previous work the author obtained
sufficient conditions on the potential V (x) which guaranteed the existence
of the wave operators, W±:= s −limU(−t)U0(t) as t →±∞. Here it is
shown that if, in addition, the associated (Schrödinger) wave operators,
W±s:= s−limei(m2I+V −Δ)te−i(m2I−Δ)t as t →∞, are complete and the Invariance
Theorem is valid then the W± are also complete and are isometries. Finally, these
results are used to show that the scattering operator, W+−1W−, is unitarily
implemented in Fock space.
