NLS scattering

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.