H1H1-scattering for systems of N-defocusing weakly coupled NLS equations in low space dimensions

We prove that the scattering operators and wave operators are well-defined in the energy space for the system of defocusing Schrödinger equations{i∂tuμ+Δuμ−∑μ,ν=1Nβμν|uν|p+1|uμ|p−1uμ=0,μ=1,…,N,(uμ(0,⋅))μ=1N=(uμ,0)μ=1N∈H1(Rd)N, with N≥2N≥2, βμν≥0βμν≥0, βμμ≠0βμμ≠0 for p>2p>2 if d=1d=1, p>1p>1 if d=2d=2 and 1≤p<21≤p<2 if d=3d=3.