Remarks on scattering for nonlinear Schrödinger equations

Abstract.

We unify two distinct methods of the global analysis for the nonlinear Schrödinger equations, namely those in the Sobolev spaces and in the weighted spaces. Thus we can deal with various sums of power nonlinearies $|u|^{p-1} u$ for 1+2/n<p<$\infty$, since the former works for $p\ge 1+4/n$, while the latter for 1+2/n<p $\le 1+4/n$. Even for a single power, our result is much simpler and slightly better than the previous ones as to restriction on the initial data. Moreover, we extend the result to the maximal regularity, thereby obtaining scattering at the lower critical value $p=1+8/(\sqrt{n^2+4n+36}+n+2)\quad \textrm{for}\quad n\ge 4$. We also show the asymptotic completeness in ${\cal F}H^1$ without smallness for $p\ge 1+8/(\sqrt{n^2+12n+4}+n-2)$ and any $n\in\mathbb{N}$.