$H^1$-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 $\begin{cases} i\partial_t u_μ+ Δu_μ- \sum_{μ,ν=1 }^N β_{μν}|u_ν|^{p+1}|u_μ|^{p-1}u_μ=0, \quad\quad μ=1,\dots,N,\\(u_μ(0,\cdot))_{μ=1}^N= (u_{μ,0})_{μ=1}^N \in H^1(\mathbb R^d)^N. \end{cases}$ with $N\geq 2$, $β_{μν} \geq 0$, $β_{μμ}\neq 0$ for $p>2$ if $d=1$, $p>1$ if $d=2$ and $1 \leq p < 2$ if $d=3$.

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