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，当原子数超过某个值（临界粒子数）时，凝聚体变得不稳定。利用一个新的不等式，严格讨论了Gross-Pitaevskii方程（非线性Schroedinger方程）的稳定性条件。2.分析了特洛伊陷阱下的玻色-爱因斯坦凝聚。特别地，我们阐明了基态的性质，基态能量降低，原子分布向中心轴移动。3.对于色散关系E k^a.我们研究了冷凝的条件。利用WKB方法，得到了磁阱的色散关系与磁阱幂律之间的关系，指出当势在r=0附近迅速减小时，即使在一维空间也会发生玻色-爱因斯坦凝聚.同时，我们还发现，a和p的值会显著地改变相变的性质。4.我们分析了凝聚体的自由高度，包括原子间的相互作用。原子波的干涉图样反映了原子间的相互作用，这在实验中是可以观察到的。5.我们发展了一维δ函数气体的统计力学。用微扰法导出了热Bethe方程的积分方程。这是贝特阿那萨茨的第一个直接证明。

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.