AI 中文总结
研究具有调和势和临界非线性、耦合常数趋于0的聚焦非线性薛定谔方程相关吉布斯测度，通过建立临界阈值，在一维和高维情况下研究频率截断测度在不同情形下的收敛性。
AI 中文摘要
本文研究与具有调和势和临界非线性的聚焦非线性薛定谔方程相关的欧几里得空间上的吉布斯测度，其耦合常数趋于0，这一问题最初由Brydges - Slade于1996年针对\(\mathbb{T}^2\)上的\(\Phi^4_2\)模型提出。在一维和高维（有径向假设）情况下，我们建立了一个临界阈值，低于此阈值频率截断测度收敛到基础高斯测度（可能有重整化的\(L^2\)截断），超临界时频率截断测度不收敛。
英文摘要
In this paper, we study the Gibbs measures on Euclidean spaces associated to the focusing nonlinear Schrödinger equation with harmonic potential and critical non linearity whose coupling constant tends to 0, a question initially posed by Brydges-Slade (1996) for the $Φ^4_2$-model on $\mathbb{T}^2$. In dimension one and in the higher dimensional cases (with radial assumption), we establish a critical threshold below which the frequency-truncated measures converge to the base Gaussian measure (possibly with a renormalized $L^2$ cut-off) while, in the supercritical regime, we prove non-convergence of the frequency-truncated measures, even up to a subsequence.
