Decomposition of global solutions of bi-laplacian Nonautonomous Schr\"odinger equations

TL;DR

This paper analyzes the long-term behavior of solutions to bi-Laplacian Schr"odinger equations with time-dependent interactions, showing they decompose into free waves and localized parts, with localization stronger in higher dimensions.

Contribution

It introduces a novel decomposition result for solutions of bi-Laplacian Schr"odinger equations with general interactions, extending understanding of their asymptotic structure.

Findings

Solutions decompose into free wave and localized components

Localization occurs in all dimensions, stronger in dimensions ≥ 9

Construction of a specialized Free Channel Wave Operator

Abstract

We study the bi-Laplacian Schr\"odinger equation with a general interaction term, which may be linear or nonlinear and is allowed to be time-dependent. We show that global solutions to such equations decompose asymptotically into a free wave and a weakly localized component in all space dimensions. Moreover, in dimensions $n \geq 9$, we prove that the weakly localized component is in fact spatially localized. The proof is based on a suitably adapted construction of the Free Channel Wave Operator, building on the method recently developed in~\cite{SW20221}.