Exchange operator formalism describes many-body integrable systems using phase-space variables involving an exchange operator that acts on any pair of particles. We establish an equivalence between models described by exchange operator formalism and the complete infinite family of parent Hamiltonians describing quantum many-body models with ground states of Jastrow form. This makes it possible to identify the invariants of motion for any model in the family and establish its integrability, even in the presence of an external potential. Using this construction we establish the integrability of the long-range Lieb-Liniger model, describing bosons in a harmonic trap and subject to contact and Coulomb interactions in one dimension. We further identify a variety of models exemplifying the integrability of Hamiltonians in this family.

中文翻译：

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Jastrow 基态的一维量子系统是可积的

交换算子形式主义描述了使用相空间变量的多体可积系统，其中涉及作用于任何粒子对的交换算子。我们在交换算子形式主义描述的模型和描述具有 Jastrow 形式基态的量子多体模型的父哈密顿量的完整无限家族之间建立了等价性。这使得识别族中任何模型的运动不变量并建立其可积性成为可能，即使在存在外部势的情况下也是如此。使用这种结构，我们建立了长程 Lieb-Liniger 模型的可积性，描述了谐波陷阱中的玻色子并在一维中受到接触和库仑相互作用。我们进一步确定了各种模型来例证哈密顿量在这个家族中的可积性。