Dynamical Localization for the One-dimensional Continuum Anderson Model in a Decaying Random Potential

We consider a one-dimensional continuum Anderson model where the potential decays in average like $$|x|^{-\alpha }$$ , $$\alpha >0$$ . We show dynamical localization for $$0<\alpha <\frac{1}{2}$$ and provide control on the decay of the eigenfunctions.