Asymptotic decoupling of bound states and radiation for time‑dependent or nonlinear Schrödinger dynamics

Determine whether, for Schrödinger equations with time‑dependent interactions or nonlinearities (including the one‑dimensional defocusing nonlinear Schrödinger equation with localized potential V(x,t) studied here), the bound‑state component and the radiative component decouple asymptotically as t → ∞. Concretely, establish that solutions split into an asymptotically orthogonal localized (bound‑state‑like) part and a freely scattering radiation part, or show that such a decoupling fails, thereby characterizing the mechanism by which slowly spreading components can obstruct decoupling.

Background

In the linear, time‑independent setting, spectral theory cleanly separates solutions into bound states (discrete spectrum) and radiation (continuous spectrum), and the radiation scatters to a free wave. This structure underlies asymptotic completeness in great generality.

For time‑dependent interactions or nonlinearities, the spectral decomposition no longer applies and the evolution of bound states and radiation need not decouple in the same way. Moreover, slowly spreading components may persist, complicating asymptotic analysis and potentially preventing a clear split into bound and radiative parts.

The paper develops exterior Morawetz and interaction Morawetz estimates for the one‑dimensional defocusing NLS with a localized potential and proves partial results such as decomposition into a free wave plus a weakly localized part with quantified spatial concentration. However, whether a full asymptotic decoupling of bound‑state‑like and radiative components holds in this broader, time‑dependent or nonlinear context remains an explicit unresolved question.