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

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}.