Wave operators to a quadratic nonlinear Klein–Gordon equation in two space dimensions

We study asymptotics around the final states of solutions to the nonlinear Klein–Gordon equations with quadratic nonlinearities in two space dimensions (∂t2−Δ+m2)u=λu2,(t,x)∈R×R2, where λ∈C. We prove that if the final states u1+∈Hqq−14−4q(R2)∩H52,1(R2)∩H12(R2),u2+∈Hqq−13−4q(R2)∩H32,1(R2)∩H11(R2), and ‖u1+‖H12+‖u2+‖H11 is sufficiently small, where 4