NLS scattering

Scattering

To improve global well-posedness for the NLS to scattering, it seems that needs $p\,$ to be $L^{2}\,$ super-critical (i.e. $p>1+4/d\,$). In this case one can obtain scattering if the data is in $L^{2}(|x|^{2}dx)\,$ (since one can then use the pseudo-conformal conservation law).

• In the $d\geq 3\,$ cases one can remove the $L^{2}(|x|^{2}dx)\,$ assumption GiVl1985 (see also Bo1998b) by exploiting Morawetz identities, approximate finite speed of propagation, and strong decay estimates (the decay of $t^{-d/2}\,$ is integrable). In this case one can even relax the $H^{1}\,$ norm to $H^{s}\,$ for some $s<1\,$ CoKeStTkTa-p8.
• For $d=1,2\,$ one can also remove the $L^{2}(|x|^{2}dx)\,$ assumption Na1999c by finding a variant of the Morawetz identity for low dimensions, together with Bourgain's induction on energy argument.

Asymptotic completeness

One can go beyond scattering and ask for asymptotic completeness and existence of the wave operators. When $p\leq 1+2/d\,$ this is not possible due to the poor decay in time in the non-linear term Bb1984, Gs1977b, Sr1989, however at $p=1+2/d\,$ one can obtain modified wave operators for data with suitable regularity, decay, and moment conditions Oz1991, GiOz1993, HaNm1998, ShiTon2004, HaNmShiTon2004. In the regime between the $L^{2}\,$ and $H^{1}\,$ critical powers the wave operators are well-defined in the energy space LnSr1978, GiVl1985, Na1999c. At the $L^{2}\,$ critical exponent $1+4/d\,$ one can define wave operators assuming that we impose an $L_{x,t}^{p}\,$ integrability condition on the solution or some smallness or localization condition on the data GiVl1979, GiVl1985, Bo1998 (see also Ts1985 for the case of finite pseudoconformal charge). Below the $L^{2}\,$ critical power one can construct wave operators on certain spaces related to the pseudo-conformal charge CaWe1992, GiOz1993, GiOzVl1994, Oz1991; see also GiVl1979, Ts1985. For $H^{s}\,$ wave operators were also constructed in Na2001. However in order to construct wave operators in spaces such as $L^{2}(|x|^{2}dx)\,$ (the space of functions with finite pseudoconformal charge) it is necessary that $p\,$ is larger than or equal to the rather unusual power

$1+8/({\sqrt {d^{2}+12d+4}}+d-2)\,$;

see NaOz2002 for further discussion.