Bose–Einstein condensates (BECs), first predicted theoretically by Bose and Einstein and finally discovered experimentally in the 1990s, continue to motivate theoretical and experimental physics work. Although experiments on BECs are carried out in bounded space domains, theoretical work in the modelling of BECs often involves solving the Gross–Pitaevskii equation on unbounded domains, as the combination of bounded domains and spatial heterogeneity render most existing analytical approaches ineffective. Motivated by a lack of theory for BECs on bounded domains, we first derive a perturbation theory for both ground and excited stationary states on a given bounded space domain, allowing us to explore the role various forms of the self-interaction, external potential and space domain have on BECs. We are able to show that the shape and curvature of a space domain strongly influence BEC structure, and may be used as control mechanisms in experiments. We next derive a non-autonomous perturbation theory to predict BEC response to temporal changes in an external potential. In certain cases, our approach can be extended to unbounded domains, and we conclude by constructing a perturbation theory for bright solitons within external potentials on unbounded domains.

中文翻译：

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有界空间域上玻色-爱因斯坦凝聚的微扰理论

玻色-爱因斯坦凝聚体 (BECs) 最初由玻色和爱因斯坦在理论上预测并最终在 1990 年代通过实验发现，继续推动理论和实验物理工作。尽管 BEC 的实验是在有界空间域中进行的，但 BEC 建模的理论工作通常涉及在无界域上求解 Gross-Pitaevskii 方程，因为有界域和空间异质性的结合使大多数现有分析方法无效。由于缺乏有界域上 BEC 的理论，我们首先推导出给定有界空间域上的基态和激发态的扰动理论，使我们能够探索各种形式的自相互作用、外部势和空间域上有 BEC。我们能够证明空间域的形状和曲率强烈影响 BEC 结构，并可用作实验中的控制机制。我们接下来推导出非自主微扰理论来预测 BEC 对外部电位时间变化的响应。在某些情况下，我们的方法可以扩展到无界域，我们通过为无界域上的外部势内的亮孤子构建微扰理论来得出结论。