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

TL;DR

This paper establishes a non-linear adiabatic theorem for a one-dimensional Landau-Pekar system, providing an approximation on the slow time scale using dispersive estimates for time-dependent Schrödinger equations.

Contribution

It introduces a novel non-linear adiabatic approximation for the 1D Landau-Pekar equations, leveraging dispersive analysis techniques.

Findings

Derived an adiabatic approximation for the system

Proved dispersive estimates for time-dependent Schrödinger equations

Validated the approximation on the slow time scale

Abstract

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.