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

非线性薛定谔方程(NLS)是多体量子系统动力学中出现的一种非线性偏微分方程组。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。这通常给我们提供了非线性偏微分方程组和量子问题之间的对应关系。后者是线性的，尽管是非对易的。它是在粒子数不固定的玻色Fock空间上提出的，NLS具有哈密顿结构，它允许我们(至少在形式上)定义在流动下不变的Gibbs测度。这种测量的构建可以追溯到20世纪70年代的构造性量子场论(Nelson，Glimm-Jaffe，Simon)，以及后来Lebowitz-Rose-Speer和McKean-Vaninsky的工作。它的不变性首先在20世纪90年代布尔格纳的开创性工作中得到了严格的表现。目前，Gibbs测度被用作研究概率低正则适定性理论的基本工具。这是因为吉布斯测量通常被支持在低正则的索博列夫空间上。我的提议的主要目的是理解吉布斯测量是如何在NLS和多体量子理论之间的对应中产生的。在量子问题中，人们使用量子吉布斯态。这些是Fock空间上的平衡态，对应于固定(正)温度下的多体哈密顿量。通过使用(经典的)Gibbs度量，人们可以类似地构造经典的Gibbs态。我们要验证的对应关系是量子Gibbs态的关联函数与经典Gibbs态的关联函数在适当定义的平均场极限下的收敛。当工作在更高维时，人们应该特别注意消除问题中出现的发散。这是通过应用灯芯排序的过程来完成的。这个过程在经典理论中是众所周知的，它有一个明显的量子类比。关于这个问题，早期的结果是由Luin-nam-Rougerie、作者与Fröhlich-Knowles-Schlein合作以及作者本人获得的。用来研究这个问题的方法来自分析，也来自概率论和统计力学。这个问题的已知结果与经典理论中的已知结果还有很大差距。也就是说，在经典理论中，可以构造具有非常奇异相互作用势的NLS的Gibbs度量。量子问题中的一个主要挑战是缺乏对易性。在这项建议中，我的目标是解决这个问题。这些技术来自分析、概率和统计力学的不同方面。一个目标是理解来自NLS分析(主要基于谐波分析)的技术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.