A system of quadratic nonlinear Klein-Gordon equations in 2d

We consider the system of quadratic nonlinear Klein-Gordon equations{∂t2v1−Δv1+m12v1=v1v2,∂t2v2−Δv2+m22v2=v12 in two space dimensions. Under the mass condition such that 2m1>m2, we construct the scattering operator in an almost natural weighted Sobolev class H1+δ,1 or the scattering problem in lower order Sobolev class H43−δ1+δ∩H1+δ, where δ>0 is small.