论文标题
Gibbs的微观推导，用于1D聚焦的立方非线性schrödinger方程
A microscopic derivation of Gibbs measures for the 1D focusing cubic nonlinear Schrödinger equation
论文作者
论文摘要
在本文中，我们对Gobbs的吉布斯度量进行了微观衍生，用于从多体量子Gibbs状态的一维圆环上进行聚焦的非线性schrödinger方程。由于我们没有在相互作用上做出任何积极的假设，因此有必要在经典环境中和量子设置中重新固定的粒子数字引入质量的截断。我们的方法基于对互动的扰动扩展，与Fröhlich，Knowles，Schlein和第二作者的先前工作类似。由于存在截断，获得的系列具有无限的收敛半径。我们处理有限，可集成和三角洲功能相互作用势的情况，没有任何符号假设。在此框架内，我们还研究了时间依赖性的相关函数。这是聚焦制度中的第一个此类已知结果。
In this paper, we give a microscopic derivation of Gibbs measures for the focusing cubic nonlinear Schrödinger equation on the one-dimensional torus from many-body quantum Gibbs states. Since we are not making any positivity assumptions on the interaction, it is necessary to introduce a truncation of the mass in the classical setting and of the rescaled particle number in the quantum setting. Our methods are based on a perturbative expansion of the interaction, similarly as in previous work of Fröhlich, Knowles, Schlein, and the second author. Due to the presence of the truncation, the obtained series have infinite radius of convergence. We treat the case of bounded, integrable, and delta function interaction potentials, without any sign assumptions. Within this framework, we also study time-dependent correlation functions. This is the first such known result in the focusing regime.
