标题： 玻色气体：一个微妙的多体问题 显示英文标题

标题： The Bose gas: A subtle many-body problem

摘要： 如今，可以在实验上研究低密度下量子力学多体系统（玻色子）基态的性质，$\rho$，因此有必要重新审视几十年前许多作者推导出的一些公式。 其中之一是能量/粒子的主要项为$4\pi a \rho$，其中$a$是两体势的散射长度。 由于玻色子相关性的微妙和特殊性质（如带电玻色子的奇怪$N^{7/5}$定律），四十年的研究未能严格证明这一合理的公式。 之前唯一找到的能量下限是由Dyson在1957年得出的，但它是正确的14分之一。 最近与J. Yngvason共同获得了正确的渐近公式，这项工作将被介绍。 将强调数学困难的原因。 一种不同的公式，直到1971年由Schick提出，适用于二维情况，这一点也将被证明是正确的。 另一个备受关注的问题是玻色-爱因斯坦凝聚的存在性，关于这一点严格已知的内容也将被讨论。 借助于证明均匀气体下限的方法论，已经成功解决了另外两个问题。 一个是与Yngvason和Seiringer一起证明了Gross-Pitaevskii方程正确描述了实验中实际使用的“陷阱”中的基态。 另一个是非常近期的证明（与Solovej一起），证明Foldy1961年关于带电粒子高密度气体的理论正确描述了其基态能量。 显示英文摘要

摘要： Now that the properties of the ground state of quantum-mechanical many-body systems (bosons) at low density, $\rho$, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. One of these is that the leading term in the energy/particle is $4\pi a \rho$ where $a$ is the scattering length of the 2-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange $N^{7/5}$ law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained jointly with J. Yngvason and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. Another problem of great interest is the existence of Bose-Einstein condensation, and what little is known about this rigorously will also be discussed. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof (with Yngvason and Seiringer) that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. The other is a very recent proof (with Solovej) that Foldy's 1961 theory of a high density gas of charged particles correctly describes its ground state energy.