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 = λ u², (t, x) ∈ R × R², where λ ∈ C. We prove that if the final states u₁⁺ ∈ H_{frac(q, q - 1)}^{4 - frac(4, q)} (R²) ∩ H^{frac(5, 2), 1} (R²) ∩ H₁² (R²),u₂⁺ ∈ H_{frac(q, q - 1)}^{3 - frac(4, q)} (R²) ∩ H^{frac(3, 2), 1} (R²) ∩ H₁¹ (R²), and {norm of matrix} u₁⁺ {norm of matrix}_H12 + {norm of matrix} u₂⁺ {norm of matrix}_H11 is sufficiently small, where 4 < q ≤ ∞, then there exists a unique global solution u ∈ C ([T, ∞) ; L² (R²)) to the nonlinear Klein-Gordon equations such that u (t) tends as t → ∞ in the L² sense to the solution u₀ (t) = u₁⁺ cos (〈 i ∇ 〉_m t) + (〈 i ∇ 〉_m^{- 1} u₂⁺) sin (〈 i ∇ 〉_m t) of the free Klein-Gordon equation.