Classification of threshold dynamics for the focusing mass-critical inhomogeneous NLS
Abstract
We study the focusing mass-critical inhomogeneous nonlinear Schrödinger equation i∂tu+Δu+|x|−b|u|(4−2b)/du=0 in dimensions d≥3. For 1/2<b<min{2,d/2}, we prove that every forward-global H1 solution with mass equal to that of the ground state Q either scatters forward or coincides with eitQ, up to phase rotation and scaling. As a consequence, every solution with nonradial H1 initial data at the ground-state mass must be global and scatter in both time directions. For 0<b≤1/2, analogous rigidity and scattering results hold under an additional finite-variance assumption. The key new ingredients in our proof are a nonradial incoming/outgoing decomposition and exterior frequency-decay estimates, which yield the kinetic-energy localization needed for a localized virial argument.
Related
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BibTeX
@misc{liu2026classification,
title = {{Classification of threshold dynamics for the focusing mass-critical inhomogeneous NLS}},
author = {Xuan Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38285},
eprint = {2610.38285},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38285},
}
RIS
TY - UNPB
TI - Classification of threshold dynamics for the focusing mass-critical inhomogeneous NLS
AU - Xuan Liu
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38285
UR - https://arxiv.org/abs/2610.38285
AB - We study the focusing mass-critical inhomogeneous nonlinear Schrödinger equation i∂_tu+Δu+|x|^−b|u|^(4−2b)/du=0 in dimensions d≥3. For 1/2<b<min{2,d/2}, we prove that every forward-global H^1 solution with mass equal to that of the ground state Q either scatters forward or coincides with e^itQ, up to phase rotation and scaling. As a consequence, every solution with nonradial H^1 initial data at the ground-state mass must be global and scatter in both time directions. For 0<b≤1/2, analogous rigidity and scattering results hold under an additional finite-variance assumption. The key new ingredients in our proof are a nonradial incoming/outgoing decomposition and exterior frequency-decay estimates, which yield the kinetic-energy localization needed for a localized virial argument.
ER -