On The large Time Asymptotics of Schr\"odinger type equations with General Data

TL;DR

This paper studies the long-time behavior of solutions to general Schrödinger equations, showing they decompose into free waves and localized parts, extending previous results to higher dimensions and non-radial data.

Contribution

It introduces a new construction of the Free Channel Wave Operator for general Schrödinger equations, broadening the scope of asymptotic analysis.

Findings

Global solutions asymptotically split into free and localized components

Extension of previous results to arbitrary dimensions

Applicable to non-radial initial data

Abstract

For the Schr\"odinger 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.