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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.