The conventional non-Hermitian but PT-symmetric three-parametric Bose–Hubbard Hamiltonian H(γ, v, c) represents a quantum system of N bosons, unitary only for parameters γ, v and c in a domain D. Its boundary ∂D contains an exceptional point of order K (EPK; K = N + 1) at c = 0 and γ = v, but even at the smallest non-vanishing parameter c ≠ ~0 the spectrum of H(v, v, c) ceases to be real, i.e. the system ceases to be observable. In this paper, the question is inverted: all of the stable, unitary and observable Bose–Hubbard quantum systems are sought which would lie close to the phenomenologically most interesting EPK-related dynamical regime. Two different families of such systems are found. Both of them are characterized by the perturbed Hamiltonians H(λ)=H(v,v,0)+λ V for which the unitarity and stability of the system is guaranteed. In the first family the number N of bosons is assumed conserved while in the second family such an assumption is relaxed. Attention is paid mainly to an anisotropy of the physical Hilbert space near the EPK extreme. We show that it is reflected by a specific, operationally realizable structure of perturbations λ V which can be considered small.

中文翻译：

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有和没有粒子数守恒的 Bose-Hubbard 例外点的酉展开

传统的非 Hermitian 但 PT 对称三参数 Bose-Hubbard Hamiltonian H(γ, v, c) 表示 N 玻色子的量子系统，仅对于域 D 中的参数 γ、v 和 c 是幺正的。其边界 ∂D在 c = 0 和 γ = v 处包含一个特殊的阶点 K (EPK; K = N + 1)，但即使在最小的非零参数 c ≠ ~0 处，H(v, v, c) 的频谱也停止是真实的，即系统不再是可观察的。在本文中，问题倒过来了：寻找所有稳定的、单一的和可观察的 Bose-Hubbard 量子系统，这些系统靠近现象学上最有趣的 EPK 相关动力学机制。发现了此类系统的两个不同系列。它们都以扰动哈密顿量 H(λ)=H(v,v,0)+λ V 为特征，保证了系统的单一性和稳定性。在第一个家族中，假设玻色子的数量 N 是守恒的，而在第二个家族中，这种假设是放松的。主要关注EPK极端附近物理希尔伯特空间的各向异性。我们表明，它是由一个特定的、可操作实现的扰动 λ V 结构反映出来的，该结构可以被认为是很小的。