We study the time decay of scattering solutions to one-dimensional Schrödinger equations and prove a weighted dispersive estimate with stronger time decay than the case of unweighted estimates for the non-resonant state. Furthermore asymptotic expansions in time of scattering solutions are given. The key of the proof is the study of the Fourier properties of the Jost functions. We improve the Fourier properties of the Jost functions obtained by D'Ancona and Fanelli .
"Dispersive estimates and asymptotic expansions for Schrödinger equations in dimension one." J. Math. Soc. Japan 63 (1) 239 - 261, January, 2011. https://doi.org/10.2969/jmsj/06310239