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A sharp criterion for well-posedness of semilinear Schrödinger equations on the real line
Abstract
We consider the Cauchy problem for semilinear Schrödinger equations on the real line. We establish a sharp necessary and sufficient condition on the nonlinearity for local well-posedness in Sobolev spaces of regularity greater than five halves. This criterion is formulated in terms of the Mizohata condition and is equivalent to explicit algebraic conditions on the nonlinearity. To prove sufficiency, we combine energy estimates with a gauge transformation adapted to the bounded primitive arising from the Mizohata condition. To prove necessity, we show that the failure of the condition yields a dense set of initial data for which no solution exists on any positive time interval.
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BibTeX
@misc{kondo2026sharp,
title = {{A sharp criterion for well-posedness of semilinear Schrödinger equations on the real line}},
author = {Toshiki Kondo and Mamoru Okamoto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21734},
eprint = {2610.21734},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21734},
}
RIS
TY - UNPB
TI - A sharp criterion for well-posedness of semilinear Schrödinger equations on the real line
AU - Toshiki Kondo
AU - Mamoru Okamoto
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21734
UR - https://arxiv.org/abs/2610.21734
AB - We consider the Cauchy problem for semilinear Schrödinger equations on the real line. We establish a sharp necessary and sufficient condition on the nonlinearity for local well-posedness in Sobolev spaces of regularity greater than five halves. This criterion is formulated in terms of the Mizohata condition and is equivalent to explicit algebraic conditions on the nonlinearity. To prove sufficiency, we combine energy estimates with a gauge transformation adapted to the bounded primitive arising from the Mizohata condition. To prove necessity, we show that the failure of the condition yields a dense set of initial data for which no solution exists on any positive time interval.
ER -