UCLA

In this thesis, we study the behavior of solutions to some semilinear Schr\"odinger equations at short and long time scales. We first consider the nonlinear Schr\"odinger equations with power-type nonlinearity in three dimensions with periodic boundary conditions. We show that this equation is locally well-posed in critically scaling Sobolev spaces $H^s(\bb{T}^3)$. We then investigate the long-time asymptotic behavior of solutions to NLS in Euclidean space with defocusing, mass-subcritical power-type and Hartree nonlinearities. We discuss the divide between the wealth of results on the scattering theory for these equations in weighted $L^2$ spaces and the paucity of analogous results in $L^2(\bb{R}^d)$. To explain this, we show that the scattering problems for these equations are well-posed in weighted $L^2$ spaces in the sense that the scattering operators attain their natural and maximal regularity. Furthermore, we show that these scattering problems are ill-posed in $L^2$ in the sense that the scattering operators cannot be extended to all of $L^2$ without losing a positive (and, in the case of Hartree, infinite) amount of regularity.