Wave operator for the system of the Dirac-Klein-Gordon equations

We prove the existence of the wave operator for the system of the massive Dirac-Klein-Gordon equations in three space dimensions xεR³∫ (∂_t+α·∇;+iMβ)+ψ= λθβψ, (∂²_t-Δ+m ²)θ=μψ*βψ, where the masses m, M>0. We prove that for the small final data φ⁺ ε(H ^3/2+μ1)⁴+(θ₁⁺, θ₂⁺) ε H^2+μ,1 with μ=5/4-5/2q and 90/37<q<6, there exists a unique global solution for system (1) with the final state conditions ∥^VD,M(-t)ψ(t)-ψ ⁺∥_H3/2+μ1→0 and ∥^VKG,m(-t) (φ(t) 〈i∇〉_m^-1∂tφ(t))- (φ₁⁺ 〈i∇〉_m ^-1φ₂⁺)∥H2+μ1→0 as t-∞.