Scattering Theory and dispersive estimates for general $1$d Charge Transfer Models
Gong Chen, Abdon Moutinho
TL;DR
This work resolves the scattering theory and dispersive analysis for general one-dimensional charge-transfer models with multiple moving potentials by refining distorted Fourier transforms and revealing structural identities that remove the prior large-velocity separation requirement. It proves the invertibility of the central dispersive map, provides a canonical decomposition of solutions into a dispersive component and moving discrete modes, and develops sharp dispersive and weighted estimates in the scattering space. The results include stability-based decompositions, asymptotic completeness for the stable regime, and the existence of wave operators, laying a rigorous foundation for asymptotic stability analyses and collision studies of multi-solitons in 1D. Overall, the paper extends the theory to arbitrary configurations of moving potentials and opens avenues for synthesis with relativistic models and multi-soliton dynamics.
Abstract
We continue our study of scattering theory and dispersive properties for one-dimensional charge transfer models, namely linear Schrödinger equations with multiple moving potentials. By the discovery of a refined structure of the construction of distorted Fourier transforms adapted to the multi-potential framework, we remove the large-velocity separation assumption imposed in [8]. This work thus completes the full scattering theory and dispersive analysis for general one-dimensional charge transfer models. These dispersive estimates provide the foundation for analyzing asymptotic stability and collision phenomena for multi-solitons in a general setting.