Transport in the One-Dimensional Schroedinger Equation

We prove a dispersive estimate for the one-dimensional Schroedinger equation, mapping between weighted $L^p$ spaces with stronger time-decay ($t^{-3\mathrm{/}2}$ versus $t^{-1\mathrm{/}2}$) than is possible on unweighted spaces. To satisfy this bound, the long-term behavior of solutions must include transport away from the origin. Our primary requirements are that $(1+\mathrm{\mid }x\mathrm{\mid }{)}^{3}V$ be integrable and $-\mathrm{\Delta }+V$ not have a resonance at zero energy. If a resonance is present (for example in the free case), similar estimates are valid after projecting away from a rank-one subspace corresponding to the resonance.