Scattering and localized states for defocusing nonlinear Schrödinger equations with potential

TL;DR

The article develops a robust framework to understand the long-time behavior of 1D defocusing NLS with a localized potential by combining exterior Morawetz and exterior interaction Morawetz estimates with an incoming/outgoing decomposition. It proves that global H^1 solutions split into a free radiation part and a weakly localized remainder, with mass concentrating at |x| ≤ t^{1/2+} and energy at |x| ≤ t^{1/3+}; these results hold for arbitrarily large initial data. In the mass-supercritical regime, it constructs a free-channel wave operator in H^1, establishing a precise asymptotic channel behavior and slow spreading for the remainder. The work advances the soliton-resolution program in time-dependent, nontrivial potentials by providing quantitative exterior-space control that compensates for the lack of strong dispersion in one dimension.

Abstract

We study the large-time behavior of global energy class ($H^1$) solutions of the one-dimensional nonlinear Schrödinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction Morawetz estimate localized to an exterior region, we prove that these solutions decompose into a free wave and a weakly localized part which is asymptotically orthogonal to any fixed free wave. We further show that the $L^2$ norm of this weakly localized part is concentrated in the region $|x| \leq t^{1/2+}$, and that the energy ($\dot{H}^1$) norm is concentrated in $|x| \leq t^{1/3+}$. Our results hold for solutions with arbitrarily large initial data.

Theorems & Definitions (41)

• Theorem 1
• Remark 1
• Remark 2
• Theorem 2
• Remark 3
• Remark 4
• Remark 5
• Lemma 3
• proof
• Theorem 4