$L^p$ Boundedness of the Scattering Wave Operators of Schr\"odinger Dynamics -- Part 2

TL;DR

This paper provides a new proof for the $L^p$ boundedness of scattering wave operators at low frequencies in Schrödinger dynamics, also enabling control over certain commutators, with implications for time-dependent potentials.

Contribution

It introduces a novel proof technique for low frequency $L^p$ boundedness and extends the analysis to time-dependent potentials.

Findings

Established $L^p$ boundedness at low frequencies

Controlled commutators involving multiplication by $|x|$

Method applicable to time-dependent potentials

Abstract

We give another proof of the $L^p$ boundedness of scattering wave operators, at the low frequency part of the data. The proof also allows the control of the commutator of multiplication by $|x|$ with the wave operator in $L^p$. The method we develop here is geared to proving similar results for \emph{time dependent potentials}, complementing previous work focused on the high frequency part.