一类具有平衡增益和损失的Calogero型量子多体问题的束缚态和关联函数
On the bound states and correlation functions of a class of Calogero-type quantum many-body problems with balanced loss and gain
摘要：
研究了具有平衡损耗和增益的多体系统的量子化。考虑了伪欧氏空间中的平移不变性模型和旋转对称模型。一个部分的运动积分的集合构造为每种类型的模型。对称不变系统的具体例子包括具有平衡损耗和增益的卡洛格罗型多体系统，其中每个粒子通过四体平方反比势加上成对的两体谐波项与其他粒子相互作用。本文还考虑了具有平衡损耗和增益的短程四体加六体平方反比势与两体谐波项的多体相互作用。一般来说，这两个模型的本征值包含量化以及连续谱。对具有连续谱的粒子进行盒归一化，可以得到包含所有粒子的完全量子化的谱和束缚态。本文讨论了在适当的Stoke楔中基态波函数的归一化问题。这两个模型的n粒子关联函数是通过将相关积分映射到随机矩阵理论中的已知结果而得到的。结果表明，具有一般多体势的旋转对称系统不具有完全实的光谱，导致不稳定的量子模。本文还研究了具有平衡损耗和增益且具有O（2，1）对称性的Hamilton系统的本征值问题。
英文摘要：
The quantization of many-body systems with balanced loss and gain is investigated. Two types of models characterized by either translational invariance or rotational symmetry under rotation in a pseudo-Euclidean space are considered. A partial set of integrals of motion are constructed for each type of model. Specific examples for the translationally invariant systems include Calogero-type many-body systems with balanced loss and gain, where each particle is interacting with other particles via four-body inverse-square potential plus pair-wise two-body harmonic terms. A many-body system interacting via short range four-body plus six-body inverse square potential with pair-wise two-body harmonic terms in presence of balanced loss and gain is also considered. In general, the eigen values of these two models contain quantized as well as continuous spectra. A completely quantized spectra and bound states involving all the particles may be obtained by employing box-normalization on the particles having continuous spectra. The normalization of the ground state wave functions in appropriate Stoke wedges is discussed. The exact n-particle correlation functions of these two models are obtained through a mapping of the relevant integrals to known results in random matrix theory. It is shown that a rotationally symmetric system with generic many-body potential does not have entirely real spectra, leading to unstable quantum modes. The eigenvalue problem of a Hamiltonian system with balanced loss and gain and admitting dynamical O(2, 1) symmetry is also considered.
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