Transport in the One-Dimensional Schroedinger Equation

Abstract

We prove a dispersive estimate for the one-dimensional Schroedinger equation, mapping between weighted $L^p$ spaces with stronger time-decay ($t^{-3/2}$ versus $t^{-1/2}$) 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 $(1+|x|)^3 V$ 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.