Dispersive analysis for one-dimensional charge transfer models

TL;DR

This paper develops a scattering theory and dispersive estimates for one-dimensional charge transfer Schrödinger models with moving potentials, advancing understanding of their long-term behavior and stability in a non-self-adjoint setting.

Contribution

It introduces a systematic approach to analyze dispersive properties of multi-potential charge transfer models with different velocities, including unstable modes.

Findings

Proves existence of wave operators and asymptotic completeness.

Establishes pointwise decay of solutions.

Handles models with threshold resonances and unstable modes.

Abstract

In this paper, we study one-dimensional linear Schr\"odinger equations with multiple moving potentials, known as transfer charge models. Focusing on the non-self-adjoint setting that arises in the study of solitons, we systematically develop the scattering theory and establish dispersive estimates under the assumption that the potentials move at significantly different velocities, even in the presence of unstable modes. In particular, we prove the existence of wave operators, asymptotic completeness, and pointwise decay of solutions, without requiring the absence of threshold resonances. Our analysis set up the fundamental work for studying the nonlinear dynamics of multi-solitons, including asymptotic stability and collisions.