AI 中文总结
研究圆上具有粗糙截断的聚焦NLS方程吉布斯测度的推导，基于弗罗利希等人的微扰展开，采用维格纳测度方法和归纳论证克服截断光滑性问题，给出聚焦五次NLS最优截断下吉布斯测度的推导。
AI 中文摘要
我们给出了在圆上具有粗糙截断的聚焦非线性薛定谔方程（NLS）的吉布斯测度的推导。这扩展了索辛格和第二作者早期的工作，他们证明了光滑截断的类似结果。我们的证明基于弗罗利希、诺尔斯、施莱因和索辛格（2017）发展的微扰展开，并为吕、南和朱（2026）最近的推导提供了另一种证明。为了证明显式项的收敛性，我们采用维格纳测度方法和归纳论证来克服截断缺乏光滑性的问题。特别地，我们给出了具有来自奥、索索埃和托洛梅奥（2022）的最优截断的聚焦五次NLS的吉布斯测度的推导。
英文摘要
We give a derivation of the Gibbs measure for the focusing nonlinear Schrödinger equation (NLS) on the circle with rough cut-off. This extends earlier work by Sohinger and the second author, which proved analogous results for smooth cut-offs. Our proof is based on the perturbative expansion developed by Fröhlich, Knowles, Schlein, and Sohinger (2017), and provides an alternative proof of the recent derivation given in Lü, Nam, and Zhu (2026). To prove convergence of the explicit terms, we employ a Wigner measure approach and an inductive argument to overcome the lack of smoothness for the cut-off. In particular, we give a derivation of the Gibbs measure for the focusing quintic NLS with the optimal cut-off from Oh, Sosoe, and Tolomeo (2022).
Comments27 pages
