Local existence, global existence, and scattering for the nonlinear Schr\"odinger equation

In this paper, we construct for every $\alpha >0$ and $\lambda \in {\mathbb C}$ a space of initial values for which there exists a local solution of the nonlinear Schr"odinger equation \begin{equation*} \begin{cases} iu_t + \Delta u + \lambda |u|^\alpha u= 0 \ u(0,x) = u_0 \end{cases} \end{equation*} on ${\mathbb{R}}^N$. Moreover, we construct for every $\alpha >\frac{2}{N}$ a class of (arbitrarily large) initial values for which there exists a global solution that scatters as $t\to \mathrm{\infty }$.