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On The large Time Asymptotics of Schrödinger type equations with General Data

Avy Soffer, Xiaoxu Wu

TL;DR

This work establishes sharp long-time asymptotics for Schrödinger-type equations with general, time-dependent and nonlinear interactions by constructing a novel Free Channel Wave Operator and proving a decomposition of global solutions into a free component plus a weakly localized remainder. The authors develop a phase-space, microlocal framework with propagation and relative-propagation estimates, enabling Cook-like arguments to pass to the limit and to handle both spatially localized and charge-transfer-type interactions in arbitrary dimensions. The results include the existence of adapted wave operators in dimensions n≥3, independence of the localization parameters, and the appearance of moving weakly localized components in the charge-transfer case, with scattering demonstrated in 3D when the nonfree part is small. Overall, the paper extends the Liu–Soffer program to general data and localized interactions, offering a robust toolkit for multichannel scattering and asymptotic completeness in time-dependent settings with potential applications to nonlinear Schrödinger equations with moving potentials.

Abstract

For the Schrödinger equation with a general interaction term, which may be linear or nonlinear, time dependent and including charge transfer potentials, we prove the global solutions are asymptotically given by the sum of a free wave and a weakly localized part. The proof is based on constructing in a new way the Free Channel Wave Operator, and further tools from the recent works \cite{Liu-Sof1,Liu-Sof2,SW2020}. This work generalizes the results of the first part of \cite{Liu-Sof1,Liu-Sof2} to arbitrary dimension, and non-radial data.

On The large Time Asymptotics of Schrödinger type equations with General Data

TL;DR

This work establishes sharp long-time asymptotics for Schrödinger-type equations with general, time-dependent and nonlinear interactions by constructing a novel Free Channel Wave Operator and proving a decomposition of global solutions into a free component plus a weakly localized remainder. The authors develop a phase-space, microlocal framework with propagation and relative-propagation estimates, enabling Cook-like arguments to pass to the limit and to handle both spatially localized and charge-transfer-type interactions in arbitrary dimensions. The results include the existence of adapted wave operators in dimensions n≥3, independence of the localization parameters, and the appearance of moving weakly localized components in the charge-transfer case, with scattering demonstrated in 3D when the nonfree part is small. Overall, the paper extends the Liu–Soffer program to general data and localized interactions, offering a robust toolkit for multichannel scattering and asymptotic completeness in time-dependent settings with potential applications to nonlinear Schrödinger equations with moving potentials.

Abstract

For the Schrödinger equation with a general interaction term, which may be linear or nonlinear, time dependent and including charge transfer potentials, we prove the global solutions are asymptotically given by the sum of a free wave and a weakly localized part. The proof is based on constructing in a new way the Free Channel Wave Operator, and further tools from the recent works \cite{Liu-Sof1,Liu-Sof2,SW2020}. This work generalizes the results of the first part of \cite{Liu-Sof1,Liu-Sof2} to arbitrary dimension, and non-radial data.
Paper Structure (22 sections, 24 theorems, 366 equations)

This paper contains 22 sections, 24 theorems, 366 equations.

Table of Contents

  1. Introduction
  2. The Scattering Problem and Results
  3. Assumptions and examples.
  4. Examples
  5. Main results.
  6. Propagation Estimate, Relative Propagation Estimate, $tT$ potentials, estimates for interaction terms and commutator estimates
  7. Propagation Estimate
  8. Relative Propagation Estimate
  9. Estimates for interaction terms
  10. Space localized $\mathcal{N}(x,t,\psi(t))$.
  11. $L^p$ potentials.
  12. Proofs of Theorem \ref{['thm1']} and Theorem \ref{['thm']}
  13. Forward/backward propagation waves
  14. Estimates for Schrödinger operators.
  15. Estimates for translated Schrödinger operators.
  16. ...and 7 more sections

Key Result

Theorem 2.4

Assume $\psi(t)$ is the solution to system SE. Let Assumptions asp: global and asp: nonlocal hold. When the space dimension $n\geq3$, for all $\alpha\in(0,1-2/n)$, the adapted free channel wave operator acting on the initial data $\psi(0)$, defined as exists in $L^2_x(\mathbb{R}^n)$. Furthermore, $\Omega_\alpha^*\psi(0)$ is independent on the choice of $\alpha$ in the following sense: for all $\a

Theorems & Definitions (68)

  • Example 2.1
  • proof
  • Example 2.2
  • proof
  • Example 2.3
  • proof
  • Theorem 2.4: $L^p$ potentials
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 58 more