Weakly localized states of one dimensional Schrodinger equations have localized energy

We study the asymptotics of the Schrödinger equation with time-dependent potential in dimension one. Assuming that the potential decays sufficiently rapidly as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$, we prove that the solution can be written as the sum of a free wave $e^{-it\mathrm{\Delta }}{u}_{+}$ and a weakly bound component $u_{\text{wb}}(t)$. Moreover, we show that the weakly bound part decomposes as $u_{\text{wb}}(t)=u_{\text{loc}}(t)+o_{{\dot{H}}^1}(1)$, where ${\mathrm{\partial }}_x{u}_{\text{loc}}(t)$ is localized near the origin uniformly in time. Since decay conditions on the potential do not preclude resonances unless $d\ge 5$, our results can be seen as a natural extension of [Terence Tao. Dynamics of Partial Differential Equations, 5(2), 2008] and [Avy Soffer, Xiaoxu Wu. arXiv:2304.04245] to the lower-dimensional case.