The theoretical foundation of Bose-Einstein condensate (BEC) was established about a century ago, and the first experimental observation of BEC was performed two decades ago. It has been interested in the dynamics of low temperature bosonic gas for interacting-systems. Assuming the initial condition to describe a BEC, i.e., a tensor product of factorized states, it has been asked that the system is in the condensation at time $t\ge0$. It has been known that the Hartree dynamics is a proper candidate for the many-body Schrödinger evolution in the mean-field regime. Many results have been obtained and a special focus is placed on the estimate of the difference between the many-body Schrödinger evolution in the mean-field regime and the corresponding Hartree dynamics. One fundamental result asserts that the optimal rate of convergence is of order $1/N$ for any fixed time $t\geq0$ with the two-body interaction potential $V\leq D(1-\Delta)$, which covers the Coulomb potential.
In this thesis, we extend this result in three different directions; the first is to relax the condition on interaction potentials by permitting more singular potential. The important lemma is proven by using Strichartz estimate. Moreover, we investigate the time dependence of the difference. To have sub-exponential bound, we use the results of time-decay estimate for small initial data. We also refine time-dependent bound for singular potential. For the time dependence we consider interaction potential $V(x)$ of type $\lambda\exp(-\mu|x|)|x|^{-\gamma}$ for $\lambda\in\mathbb{R}$, $\mu\geq0,$and $0<\gamma<3/2$, which covers the Coulomb and Yukawa interaction. The last is to extend the result for many components BEC, i.e., a system of $p$ components of bosons, each of which consists of $N_{1},N_{2},\ldots,N_{p}$ particles, respectively. The bosons are in three dimensions with interactions via a generalized interaction potential which includes the Coulomb interaction. We show that the optimal rate of convergence.
보즈-아인슈타인 응축과 관련하여 저온에서 상호작용하는 보존 입자의 동역학에 관한 관심이 계속 해서 이어져오고 있다. 본 연구에서는 한 번 응축된 상태가 시간이 지나도 계속해서 응축 상태인가에 관한 연구를 하였다. 슈뢰딩거 방정식을 따르는 보존 입자들은 응축 평균장 이론에 따라 하트리동역학을 따르며 이때의 수렴성에 관련하여 많은 결과들이 있다. 쿨롬 포텐셜에서는 최적수렴성이 1/N 정도라고 알려져있다. 본 학위논문에서 우리는 위 결과들을 세 가지 다른 방향으로 일반화하였다. 첫번째로 상호작용 포텐셜에 관한 조건을 완화하여, 조금 더 특이한 포텐셜에서 또한 동일한 수렴성을 얻었다. 두번째로 수렴성의 시간 의존성에 관한 조사를 하였다. 마지막으로 다요소 조건하에서의 BEC를 살펴보았다. 각 경우에 대해 최적 수렴성 1/N 을 얻었다.
