We study the derivation of the Gibbs measure for the nonlinear Schrödinger (NLS) equation from many-body quantum thermal states in the mean-field limit. In this paper, we consider the nonlocal NLS with defocusing and unbounded $L^p$ interaction potentials on $\mathbb{T}^d$ for $d=1,2,3$. This extends the author’s earlier joint work with Fröhlich et al. [ 45], where the regime of defocusing and bounded interaction potentials was considered. When $d=1$, we give an alternative proof of a result previously obtained by Lewin et al. [ 69]. Our proof is based on a perturbative expansion in the interaction. When $d=1$, the thermal state is the grand canonical ensemble. As in [ 45], when $d=2,3$, the thermal state is a modified grand canonical ensemble, which allows us to estimate the remainder term in the expansion. The terms in the expansion are analysed using a graphical representation and are resummed by using Borel summation. By this method, we are able to prove the result for the optimal range of $p$ and obtain the full range of defocusing interaction potentials, which were studied in the classical setting when $d=2,3$ in the work of Bourgain [ 15].

中文翻译：

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具有无限相互作用势的非线性薛定谔方程吉布斯测度的微观推导

我们研究了从平均场极限中的多体量子热态推导非线性薛定谔 (NLS) 方程的吉布斯测度。在本文中，我们考虑在 $\mathbb{T}^d$ 上对于 $d=1,2,3$ 具有散焦和无界 $L^p$ 交互潜力的非局部 NLS。这扩展了作者早期与 Fröhlich 等人的合作。[45]，其中考虑了散焦和有界相互作用势的制度。当 $d=1$ 时，我们给出 Lewin 等人先前获得的结果的替代证明。[69]。我们的证明是基于相互作用中的微扰扩展。当$d=1$ 时，热态是大正则系综。如在[45]中，当$d=2,3$时，热态是一个修正的大正则系综，它允许我们估计展开中的余项。展开式中的项使用图形表示进行分析，并通过使用 Borel 求和来恢复。通过这种方法，我们能够证明 $p$ 的最佳范围的结果，并获得全范围的散焦相互作用势，这在 Bourgain 的工作中在 $d=2,3$ 的经典设置中进行了研究[ 15]。