Energy Scattering for Nonlinear Klein–Gordon and Schrödinger Equations in Spatial Dimensions 1 and 2

We show that when n=1 and 2, the scattering operators are well-defined in the whole energy space for nonlinear Klein–Gordon and Schrödinger equations in R1+n with nonlinearity |u|p−1 u, p>1+4/n. Such results have been known only for n⩾3.