$H^1$-scattering for systems of $N$-defocusing weakly coupled NLS equations in low space dimensions

Abstract

We prove that the scattering operators and wave operators are well-defined in the energy space for the system of defocusing Schr\"odinger equations $$ \begin{cases} i\partial_t u_\mu + \Delta u_\mu - \sum_{\mu,\nu=1 }^N \beta_{\mu\nu}|u_\nu|^{p+1}|u_\mu|^{p-1}u_\mu=0, \quad\quad \mu=1,\dots,N,\\(u_\mu(0,\cdot))_{\mu=1}^N= (u_{\mu,0})_{\mu=1}^N \in H^1(\mathbb R^d)^N. \end{cases} $$ with $N\geq 2$, $\beta_{\mu\nu} \geq 0$, $\beta_{\mu\mu}\neq 0$ for $p>2 $ if $d=1$, $p>1 $ if $d=2$ and $ 1 \leq p < 2$ if $d=3$.