关于均匀玻色气体能量-动量谱的下确界
On the infimum of the energy-momentum spectrum of a homogeneous Bose gas
摘要：
本文研究了零温下具有周期性边界条件的立方盒子中的二次量子化均匀玻色气体。讨论了玻色气体的能量动量谱及其物理意义。我们回顾了各种严格的和启发式的结果，以及公开的关于它的性质。我们的主要目的是说服读者，包括那些主要是数学背景，这个问题有许多有趣的问题，严格的研究。特别地，我们研究了由压缩态上单粒子激发的期望值给出的固定总动量$\kk$的能量下确界的上界。这个界限可以被看作是著名的Bogoliubov方法的严格版本。我们表明，这种方法似乎会导致（非物理的）能隙。压缩态的变分问题可以作为微扰法计算激发谱的预备步骤。玻色气体的微扰方法似乎（至少在原则上）优于基于$c$数替换的常用方法。
英文摘要：
We consider second quantized homogeneous Bose gas in a large cubic box with periodic boundary conditions, at zero temperature. We discuss the energy-momentum spectrum of the Bose gas and its physical significance. We review various rigorous and heuristic results as well as open conjectures about its properties. Our main aim is to convince the readers, including those with mainly mathematical background, that this subject has many interesting problems for rigorous research. In particular, we investigate the upper bound on the infimum of the energy for a fixed total momentum $\kk$ given by the expectation value of one-particle excitations over a squeezed states. This bound can be viewed as a rigorous version of the famous Bogoliubov method. We show that this approach seems to lead to a (non-physical) energy gap. The variational problem involving squeezed states can serve as the preparatory step in a perturbative approach that should be useful in computing excitation spectrum. This version of a perturbative approach to the Bose gas seems (at least in principle) superior to the commonly used approach based on the $c$-number substitution.
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