Scattering theory for temporally inhomogeneous Hamilton equations in Hilbert space (Q1100635)

In the scattering theory for the equation \(y'+P(t)y=0\) in Banach space X it is known that, for a perturbation which is summable in time, the scattering operator is everywhere defined in the space of the generalized solutions and onto. We prove here that a stronger notion of scattering operator, defined in the space of the true solutions, is everywhere defined and onto, provided that X is reflexive and that P satisfies the following property: if \(\hat P\) denotes the average operator \(\hat P(t)=\int^{t+1}_{t}P(s)ds,\) then the Cauchy problem for the equation \(y'+\hat P(t)y=0\) is uniformly well posed on the real line. This result is applied to the Hamilton equation in Hilbert space \((Jx')=H(t)x,\) generalizing those of the author [Arch. Ration. Mech. Anal. 86, 147-180 (1984; Zbl 0563.35041)] for the abstract wave equation \(u''+A(t)u=0\).