$H^1$-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 Schrödinger equations $$\{\begin{array}{l}i{\partial }_t{u}_{\mu }+\Delta u_{\mu }-{\sum }_{\mu ,\nu =1}^N{\beta }_{\mu \nu }|u_{\nu }{|}^{p+1}|u_{\mu }{|}^{p-1}{u}_{\mu }=0,\mu =1,\dots ,N,\\ (u_{\mu }(0,\cdot ){)}_{\mu =1}^N=(u_{\mu ,0}{)}_{\mu =1}^N\in H^1({\mathbb{R}}^d{)}^N.\end{array}$$ with $N\ge 2$, ${\beta }_{\mu \nu }\ge 0$, ${\beta }_{\mu \mu }\ne 0$ for $p>2$ if $d=1$, $p>1$ if $d=2$ and $1\le p<2$ if $d=3$.