A non-linear adiabatic theorem for the one-dimensional Landau-Pekar equations

We discuss a one-dimensional version of the Landau-Pekar equations, which are a system of coupled differential equations with two different time scales. We derive an approximation on the slow time scale in the spirit of a non-linear adiabatic theorem. Dispersive estimates for solutions of the Schr\"odinger equation with time-dependent potential are a key technical ingredient in our proof.