Nonlinear analysis of Bose-Einstein Condensates

1.在原子间的有效相互作用是吸引的情况下，当原子数目超过某一值(临界粒子数)时，凝聚体变得不稳定。利用一个新的不等式，我们严格地讨论了Gross-Pitaevskii方程(非线性薛定谔方程)的稳定性条件。2.分析了玻色-爱因斯坦凝聚体在螺线陷阱下的稳定性。特别是，我们阐明了基态的性质，基态能量降低，原子的分布向中轴移动。3.对于色散关系E~k^A，我们研究了凝聚的条件。利用WKB方法，我们找到了磁囚禁V~r^p的色散关系和幂定律之间的关系，指出当势在r=0附近迅速下降时，甚至在一维范围内也会发生玻色-爱因斯坦凝聚。4.分析了考虑原子相互作用的凝聚态的自由高度。原子波的干涉图样反映了实验上可以观察到的相互作用。5.我们发展了一维Delta函数气体的统计力学。用摄动法导出了热Bethe ansatz方程的积分方程。这是Bethe Anasatz的第一个直接证据。

1.In the case that the effective interaction between atoms is attractive, the condensate becomes unstable when the number of atoms exceeds some value (critical particle number). Using a new inequality, we discussed rigorously the stability condition of Gross-Pitaevskii equation (Nonlinear Schroedinger equation).2.We analyzed the Bose-Einstein condensate under a troidal trap. In Particular, we clarified the ground state properties the ground state energy decreases and the distribution of atoms is shifted to the central axis.3.For the dispersion relation E〜k^a. we investigated the condition of the condensation. By using the WKB method, we found the relation between the dispersion relation and the power-law of the magnetic trap V〜r^p. When the potential decreases rapidly around r=0, we pointed out that the Bose-Einstein condensation takes place even in one dimension. Also, we showed that the values of a and p change drastically the properties of the transition.4.We analyzed the free tall of the condensate including the atomic interactions. Interference patterns of atomic waves reflect the interactions, which are observable in experiments.5.We developed a statistical mechanics of one-dimensional delta function gas. By the peturbational method, we derived the integral equation of thermal Bethe ansatz equation. This is a fisrt direct proof of the Bethe anasatz.