Transport in the one-dimensional Schrödinger equation

We prove a dispersive estimate for the Schrödinger equation on the real line, mapping between weighted $L^p$ spaces with stronger time-decay ( $\vert t\vert^{-\frac32}$ versus $\vert t\vert^{-\frac12}$) 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 $\langle x\rangle^{3}V$ be integrable and $-\Delta + V$ 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.