Scattering operator for nonlinear Klein–Gordon equations in higher space dimensions

We prove the existence of the scattering operator in the neighborhood of the origin in the weighted Sobolev space Hβ,1Hβ,1 with β=max(32,1+2n) for the nonlinear Klein–Gordon equation with a power nonlinearityutt−Δu+u=μ|u|σ−1u,(t,x)∈R×Rn, where 1+4n+2<σ<1+4n for n⩾3n⩾3, μ∈Cμ∈C.