L^p Boundedness of the Scattering Wave Operators of Schroedinger Dynamics with Time-dependent Potentials and applications

TL;DR

This paper proves the $L^p$ boundedness of wave operators for Schrödinger equations with time-dependent potentials, enabling new results in scattering theory and nonlinear dispersive equations.

Contribution

It introduces new cancellation lemmas to establish $L^p$ boundedness of wave operators for time-dependent potentials, advancing scattering theory and nonlinear analysis.

Findings

Proves $L^p$ boundedness of wave operators with time-dependent potentials.

Establishes global existence and scattering for certain nonlinear Schrödinger equations.

Demonstrates existence of free channel wave operators in $L^p$ spaces for $p>p_c(n)$.

Abstract

This paper establishes the $L^p$ boundedness of wave operators for linear Schr\"odinger equations in $\mathbb{R}^3$ with time-dependent potentials. The approach to the proof is based on new cancellation lemmas. As a typical application based on this method, combined with Strichartz estimates is the existence and scattering for nonlinear dispersive equations. For example, we prove global existence and uniform boundedness in $L^{\infty}$, for a class of Hartree nonlinear Schr\"odinger equations in $L^2(\mathbb{R}^3),$ allowing the presence of solitons. We also prove the existence of free channel wave operators in $L^p(\mathbb{R}^n)$ for $p>p_c(n)$, with $p_c(3)=6$.