Sharp stability near sums of ground states for fractional Schrödinger equations
Abstract
We study quantitative stability for the fractional Schrödinger equation (−Δ)s u+u−|u|αu=0 near finite sums of widely separated positive ground states. For every n≥1, 0<s<1, and Sobolev-subcritical exponent α>0, we estimate the Hs distance to the family of such sums in terms of the H−s norm of the equation's residual. The optimal rate changes at α=n/[2(n+2s)], with a logarithmic correction at the threshold. Above the threshold the rate is t(n+2s)/(n+2s+1); below it the exponent is [(1+α)(n+2s)−n/2]/(n+2s+1). The proof combines uniform invertibility away from the translation modes with precise interaction estimates and a weighted correction of the approximate configuration. This correction resolves the translation interactions even when the nonlinearity has a small exponent. We also construct nonnegative configurations that attain these rates. The estimates quantify the effect of the algebraic decay of fractional ground states on stability.
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BibTeX
@misc{chen2026sharp,
title = {{Sharp stability near sums of ground states for fractional Schrödinger equations}},
author = {Hua Chen and Yun-Lu Fan and Xin Liao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11204},
eprint = {2610.11204},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11204},
}
RIS
TY - UNPB
TI - Sharp stability near sums of ground states for fractional Schrödinger equations
AU - Hua Chen
AU - Yun-Lu Fan
AU - Xin Liao
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11204
UR - https://arxiv.org/abs/2610.11204
AB - We study quantitative stability for the fractional Schrödinger equation (−Δ)^s u+u−|u|^αu=0 near finite sums of widely separated positive ground states. For every n≥1, 0<s<1, and Sobolev-subcritical exponent α>0, we estimate the H^s distance to the family of such sums in terms of the H^−s norm of the equation's residual. The optimal rate changes at α=n/[2(n+2s)], with a logarithmic correction at the threshold. Above the threshold the rate is t^(n+2s)/(n+2s+1); below it the exponent is [(1+α)(n+2s)−n/2]/(n+2s+1). The proof combines uniform invertibility away from the translation modes with precise interaction estimates and a weighted correction of the approximate configuration. This correction resolves the translation interactions even when the nonlinearity has a small exponent. We also construct nonnegative configurations that attain these rates. The estimates quantify the effect of the algebraic decay of fractional ground states on stability.
ER -