We introduce a model of trapped bosons with contact interactions as well as Coulomb repulsion or gravitational attraction in one spatial dimension. We find the exact ground state energy and many-body wave function. The density profile and the pair correlation function are sampled using Monte Carlo method and show a rich variety of regimes with crossovers between them. Strong attraction leads to a trapped McGuire quantum soliton. Weak repulsion results in an incompressible Laughlin-like fluid with flat density, well reproduced by a Gross-Pitaevskii equation with long-range interactions. Stronger repulsion induces Friedel oscillations and the eventual formation of a Wigner crystal.

中文翻译：

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具有短程和长程相互作用的一维俘获玻色子的精确可解系统

我们介绍了在一个空间维度上具有接触相互作用以及库仑排斥或引力吸引的玻色子的模型。我们找到确切的基态能量和多体波函数。使用蒙特卡洛方法对密度分布图和对相关函数进行采样，结果表明它们之间有交叉的机制非常丰富。强烈的吸引力导致了被困的麦奎尔量子孤子。弱斥力会导致不可压缩的Laughlin状流体具有平坦的密度，并可以通过具有长程相互作用的Gross-Pitaevskii方程很好地重现。较强的排斥作用会引起Friedel振荡，并最终形成Wigner晶体。