Gibbs measures for nonlinear Schrodinger equations and many-body quantum mechanics

非线性薛定谔方程（NLS）是一个非线性偏微分方程，它出现在多体量子系统的动力学中。这种对应的一个例子可以在玻色-爱因斯坦凝聚现象中看到。NLS的解对应于玻色-爱因斯坦凝聚。这通常给我们一个非线性偏微分方程和量子问题之间的对应。后者是线性的，尽管是非交换的。它是在玻色Fock空间上提出的，在这个空间中粒子的数目是不固定的。NLS具有一个Hamilton结构，它允许我们（至少在形式上）定义一个吉布斯测度，它在流动下是不变的。这种测度的建立可以追溯到20世纪70年代的构造性量子场论（纳尔逊、格里姆-贾菲和西蒙的工作），以及后来的莱博维茨-罗斯-斯佩尔和麦金恩-瓦宁斯基的工作。它的不变性首先在布尔甘在20世纪90年代的开创性工作中得到严格证明。吉布斯测度是研究概率低正则适定性理论的基本工具。这是因为吉布斯测度通常在低正则性的Sobolev空间上得到支持。我的建议的主要目标是理解吉布斯测度是如何在NLS和多体量子理论之间的对应中出现的。在量子问题中，人们处理量子吉布斯态。这些都是平衡态的福克空间对应的多体哈密顿量在一个固定的（正）温度。通过使用（经典）吉布斯测度，人们可以类似地构造经典吉布斯态。我们要验证的对应关系是量子吉布斯态的相关函数在适当定义的平均场极限下与经典吉布斯态的相关函数的收敛性。当在高维中工作时，应该特别注意消除问题中出现的发散。这是通过应用Wick排序的过程来完成的。这个过程是众所周知的经典理论，它有一个明确的量子模拟。早期的结果，这个问题得到了列文南Rougerie，作者在合作与Fröhlich-Knowles-Schlein，并由作者本人。用于研究这个问题的方法来自分析，但也来自概率和统计力学。在这个问题中，我们所知道的和经典理论中所知道的仍然有很大的差距。也就是说，在经典理论中，对于具有非常奇异的相互作用势的NLS，构造Gibbs测度是可能的。量子问题的一个主要挑战是缺乏交换性。在这份建议中，我的目的是解决这个问题。这些技术来自分析、概率和统计力学的不同方面。一个目标是理解NLS分析技术（主要基于谐波分析）和量子场论方法之间的联系。

The nonlinear Schrödinger equation (NLS) is a nonlinear PDE that arises in the dynamics of many-body quantum systems. An instance of this correspondence can be seen in the phenomenon of Bose-Einstein condensation. The solution of the NLS corresponds to the Bose-Einstein condensate. This in general gives us a correspondence between a nonlinear PDE and a quantum problem. The latter is linear, albeit non-commutative. It is posed on the bosonic Fock space, in which the number of particles is not fixed.The NLS possesses a Hamiltonian structure, that allows us to (at least formally) define a Gibbs measure, which is invariant under the flow. The construction of such a measure dates from the constructive quantum field theory in the 1970s (the work of Nelson, Glimm-Jaffe, Simon), and later work of Lebowitz-Rose-Speer and McKean-Vaninsky. Its invariance was first rigorously shown in the pioneering work of Bourgain in the 1990s. Today, Gibbs measures are used as a fundamental tool in the study of probabilistic low-regularity well-posedness theory. This is due to the fact that Gibbs measures are typically supported on low-regularity Sobolev spaces.The main goal of my proposal is to understand how Gibbs measures arise in the correspondence between the NLS and many-body quantum theory. In the quantum problem, one works with quantum Gibbs states. These are equilibrium states on Fock space corresponding to the many-body Hamiltonian at a fixed (positive) temperature. By using the (classical) Gibbs measure, one can similarly construct the classical Gibbs states. The correspondence that we want to verify is the convergence of correlation functions of the quantum Gibbs state to those of the classical Gibbs state in an appropriately defined mean-field limit.When working in higher dimensions, one should take special care to eliminate the divergences that arise in the problem. This is done by applying the procedure of Wick-ordering. This procedure is well-known in the classical theory and it has a clear quantum analogue.Earlier results on this problem were obtained by Lewin-Nam-Rougerie, by the author in collaboration with Fröhlich-Knowles-Schlein, and by the author himself. The methods used to study the problem came from analysis, but also from probability, and statistical mechanics. There is still a substantial gap with what is known in this problem and what is known in the classical theory. Namely, in the classical theory it is possible to construct Gibbs measures for the NLS with very singular interaction potentials. A major challenge in the quantum problem is the lack of commutativity. In this proposal, I aim to tackle this problem. The techniques come from different aspects of analysis, probability, and statistical mechanics. One goal would be to understand connections between techniques from the analysis of the NLS (which are primarily based on harmonic analysis) and the methods of quantum field theory.