The good commutator approach to global asymptotics for the Schrödinger equation with variable coefficients
Abstract
We present a robust physical-space approach to establish time decay and global asymptotics of solutions to variable-coefficient Schrödinger equations in (3+1)-dimensions. As an immediate nonlinear application, we obtain new small data global existence and asymptotics results for quasilinear Schrödinger equations with cubic, Hamiltonian nonlinearity, variable coefficients in their linear part, and possibly outside obstacles, even in the presence of trapped bicharacteristics (provided that they are suitably unstable). Our approach relies on three primary ingredients. First is the concept of a good commutator, which extends Klainerman's classical commuting vector field method, and develops upon earlier approaches of Cuccagna--Georgiev--Visciglia and Rodnianski--Tao in the variable-coefficient case, and of Ifrim--Tataru, Ifrim--Koch--Tataru in the nonlinear case. Second, we apply Ifrim and Tataru's testing-by-wave-packets method, which is key for obtaining global asymptotics and handling sharp decay assumptions on the coefficients. Finally, we introduce a systematic technique for analyzing the control provided by the good commutator, termed two-scale elliptic analysis. Together, these techniques significantly broaden the applicability of physical-space methods to a wider class of variable-coefficient nonlinear problems.
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BibTeX
@misc{oh2026good,
title = {{The good commutator approach to global asymptotics for the Schrödinger equation with variable coefficients}},
author = {Sung-Jin Oh and Federico Pasqualotto and Ning Tang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.00268},
eprint = {2608.00268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.00268},
}
RIS
TY - UNPB
TI - The good commutator approach to global asymptotics for the Schrödinger equation with variable coefficients
AU - Sung-Jin Oh
AU - Federico Pasqualotto
AU - Ning Tang
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2608.00268
UR - https://arxiv.org/abs/2608.00268
AB - We present a robust physical-space approach to establish time decay and global asymptotics of solutions to variable-coefficient Schrödinger equations in (3+1)-dimensions. As an immediate nonlinear application, we obtain new small data global existence and asymptotics results for quasilinear Schrödinger equations with cubic, Hamiltonian nonlinearity, variable coefficients in their linear part, and possibly outside obstacles, even in the presence of trapped bicharacteristics (provided that they are suitably unstable). Our approach relies on three primary ingredients. First is the concept of a good commutator, which extends Klainerman's classical commuting vector field method, and develops upon earlier approaches of Cuccagna--Georgiev--Visciglia and Rodnianski--Tao in the variable-coefficient case, and of Ifrim--Tataru, Ifrim--Koch--Tataru in the nonlinear case. Second, we apply Ifrim and Tataru's testing-by-wave-packets method, which is key for obtaining global asymptotics and handling sharp decay assumptions on the coefficients. Finally, we introduce a systematic technique for analyzing the control provided by the good commutator, termed two-scale elliptic analysis. Together, these techniques significantly broaden the applicability of physical-space methods to a wider class of variable-coefficient nonlinear problems.
ER -