ℒ^{𝓅} boundedness of the scattering wave operators of Schrödinger dynamics with time-dependent potentials and applications–part I

Soffer, Avy; Wu, Xiaoxu

doi:10.1090/tran/9414

$\mathcal {L}^p$ boundedness of the scattering wave operators of Schrödinger dynamics with time-dependent potentials and applications–part I
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by Avy Soffer and Xiaoxu Wu;

Trans. Amer. Math. Soc. 378 (2025), 4437-4507

DOI: https://doi.org/10.1090/tran/9414

Published electronically: March 25, 2025

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Abstract:

This paper establishes the $\mathcal {L}^p$ boundedness of wave operators localized at high-frequency 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 $\mathcal {L}^{\infty }$, for a class of Hartree nonlinear Schrödinger equations in $\mathcal {L}^2(\mathbb {R}^3)$, allowing the presence of solitons. We also prove the existence of free channel wave operators in $\mathcal {L}^p(\mathbb {R}^3)$ for $p>6$.

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$\mathcal {L}^p$ boundedness of the scattering wave operators of Schrödinger dynamics with time-dependent potentials and applications–part I
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Abstract: