Vol. 31, No. 1, 1969

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The asymptotic behavior of the Klein-Gordon equation with external potential. II

John Martin Chadam

Vol. 31 (1969), No. 1, 19–31

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)t ei(m2IΔ)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.

Mathematical Subject Classification
Primary: 81.47
Received: 19 August 1968
Published: 1 October 1969
John Martin Chadam