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

TL;DR

This work studies the $L^p$ boundedness of wave operators for the non-autonomous Schrödinger evolution in $\mathbb{R}^3$ with time-dependent potentials $V(x,t)$. Central to the approach are the integrated tT (I_t) cancellation lemmas, including the improved cancellation lemma (ICL), which yield uniform $L^p$ bounds on high-frequency components of the wave operators and enable free channel wave operators to exist in $L^p$ for $p>6$. The results cover a broad class of time-dependent potentials (including Mikhlin-type in time, self-similar, and moving/quench models) and yield Strichartz-type estimates for the high-frequency dynamics, with explicit nonlinear applications such as global well-posedness and $L^{\infty}$-boundedness for Hartree-type NLS, even in the presence of solitons. Collectively, the paper extends scattering and dispersive techniques to nonautonomous Hamiltonians, providing a robust framework for high-frequency analysis and nonlinear scattering in time-dependent settings.

Abstract

This paper establishes the $L^p$ boundedness of wave operators for linear Schrödinger 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 Schrödinger 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 $p_c(3)=6$.

Theorems & Definitions (137)

• Theorem 1.1
• Remark 1
• Remark 2
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 3
• Theorem 1.5
• Theorem 1.6: Existence of free channel wave operator in $\mathcal{L}^p_x$
• Remark 4