分数阶薛定谔方程基态和附近的尖锐稳定性
Sharp stability near sums of ground states for fractional Schrödinger equations
- School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
- MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院教育部重点实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对分数阶薛定谔方程，定量分析其在有限个相距甚远的正基态之和附近的稳定性，推导了最优速率及阈值条件，构造了达到速率的非负构型，量化了分数阶基态代数衰减对稳定性的影响。
AI中文摘要：
我们研究分数阶薛定谔方程 $(-\Delta)^s u+u-|u|^\alpha u=0$ 在有限个相距甚远的正基态之和附近的定量稳定性。对任意 $n\ge1$、$0<s<1$ 及索伯列夫次临界指数 $\alpha>0$，我们根据方程残差的 $H^{-s}$ 范数估计了此类基态和族的 $H^s$ 距离。最优速率在 $\alpha=n/[2(n+2s)]$ 处发生变化，阈值处存在对数修正：阈值以上速率为 $t^{(n+2s)/(n+2s+1)}$，阈值以下指数为 $[(1+\alpha)(n+2s)-n/2]/(n+2s+1)$。证明结合了远离平移模的一致可逆性、精确相互作用估计以及近似构型的加权修正，该修正即使在非线性指数较小时也能解决平移相互作用问题。我们还构造了达到这些速率的非负构型，这些估计量化了分数阶基态的代数衰减对稳定性的影响。
英文摘要：
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 $n\ge1$, $0<s<1$, 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.
