A Microscopic Derivation of Gibbs Measures for Nonlinear Schrödinger Equations with Unbounded Interaction Potentials
A Microscopic Derivation of Gibbs Measures for Nonlinear Schrödinger Equations with Unbounded Interaction Potentials
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DOI：
10.1093/imrn/rnab132
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发表时间：
2019-04
影响因子：
1
通讯作者：
Vedran Sohinger
中科院分区：
文献类型：
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作者：
Vedran Sohinger
研究了在平均场极限下，由多体量子热态导出非线性薛定谔（NLS）方程的吉布斯测度。本文考虑了$\mathbb{T}^d$（d= 1，2，3 $）上具有散焦和无界相互作用势的非局部非线性最小二乘问题.这扩展了作者与Fröhlich等人的早期联合工作。[ 45]，其中考虑了散焦和有界相互作用势的机制。当d=1时，给出Lewin等人[ 69]的一个结果的另一种证明.我们的证明是基于相互作用中的微扰展开。当d=1时，热态是巨正则系综.与[ 45]一样，当$d= 2，3 $时，热态是修改的巨正则系综，这允许我们估计展开中的余项。在扩展中的条款进行了分析，使用图形表示，并通过使用Borel求和。利用这种方法，我们能够证明p$的最佳范围的结果，并得到全范围的离焦相互作用势，这是Bourgain [ 15]在经典设置中当d= 2，3 $时所研究的。
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].