Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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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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

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 $p>6$.
References
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Bibliographic Information
  • Avy Soffer
  • Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
  • MR Author ID: 198594
  • Email: soffer@math.rutgers.edu
  • Xiaoxu Wu
  • Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
  • ORCID: 0000-0003-2972-8153
  • Email: xw292@math.rutgers.edu
  • Received by editor(s): March 4, 2022
  • Received by editor(s) in revised form: January 20, 2023, and December 16, 2024
  • Published electronically: March 25, 2025
  • Additional Notes: The first author was supported in part by NSF grant DMS-1600749 and by NSFC11671163
    The second author is supported by ARC-FL220100072
  • © Copyright 2025 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 378 (2025), 4437-4507
  • MSC (2020): Primary 35P25, 35Q55, 47A40
  • DOI: https://doi.org/10.1090/tran/9414