量子气体中的激发谱和准粒子。严格的方法
Excitation spectrum and quasiparticles in quantum gases. A rigorous approach
摘要：
本论文致力于相互作用量子气体的严格研究。感兴趣的主要对象是激发谱和准粒子的密切相关的概念。这项工作的直接动机是提出关于这两个概念的光谱观点。本文第一部分讨论了激发谱和准粒子的概念。我们提供了一个物理动机的概述，并在此基础上，我们提出了一个频谱和哈密顿为基础的方法对这些条款。在此基础上，我们制定了与这些概念相关的定义和命题。在第二部分中，我们回顾了Bogoliubov和Hartree-Fock-Bogoliubov近似，在物理学文献中，它们被用来获得准粒子图像。我们展示了这两种方法如何适合一个通用的计划，使我们能够在一个更一般的设置在一个准粒子图片。该方案基于在所谓的高斯态上的哈密顿量的最小化。它的抽象形式是Beliaev定理的内容。在最后一部分，我们提出了一个严格的结果有关的理由Bogoliubov近似。这个理由采用了平均场和无限体积极限的概念。我们发现，对于大量的粒子，一个大的体积和足够高的密度，低洼的能量动量谱的均匀玻色气体是很好地描述了Bogoliubov近似。这一结果以定理的形式表示出来，可以看作是本文的主要结果。
英文摘要：
This thesis is devoted to a rigorous study of interacting quantum gases. The main objects of interest are the closely related concepts of excitation spectrum and quasiparticles. The immediate motivation of this work is to propose a spectral point of view concerning these two concepts. In the first part of this thesis we discuss the concepts of excitation spectrum and quasiparticles. We provide an overview of physical motivations and based on that we propose a spectral and Hamiltonian-based approach towards these terms. Based on that, we formulate definitions and propositions related to these concepts. In the second part we recall the Bogoliubov and Hartree-Fock-Bogoliubov approximations, which in the physics literature are used to obtain the quasiparticle picture. We show how these two approaches fit into a universal scheme which allows us to arrive at a quasiparticle picture in a more general setup. This scheme is based on the minimization of Hamiltonians over the so-called Gaussian states. Its abstract formulation is the content of Beliaev's Theorem. In the last part we present a rigorous result concerning the justification of the Bogoliubov approximation. This justification employs the concept of the mean-field and infinite-volume limit. We show that for a large number of particles, a large volume and a sufficiently high density, the low-lying energy-momentum spectrum of the homogeneous Bose gas is well described by the Bogoliubov approximation. This result, which is formulated in the form of a theorem, can be seen as the main result of this thesis.
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