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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 ].

A Microscopic Derivation of Gibbs Measures for Nonlinear Schrödinger Equations with Unbounded Interaction Potentials