A new kind of symmetry behaviour introduced as partial \(\mathcal {P}\mathcal {T}\)-symmetry(henceforth \(\partial _{\mathcal {P}\mathcal {T}}\)) is investigated in a typical Fock space setting. The said Fock space is understood as a Reproducing Kernel Hilbert Space (RKHS). The same kind of symmetry is analysed for a non-hermitian Bose-Hubbard type Hamiltonian (involving two boson operators) along with its eigenstates. The phenomenon of symmetry breaking has also been considered.

中文翻译：

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将部分 PT $\mathcal {P}\mathcal {T}$ 对称性理解为再现核希尔伯特空间中的加权复合共轭：非厄米玻色哈伯德型哈密顿量在 Fock 空间中的应用

一种新的对称行为被引入为部分\(\mathcal {P}\mathcal {T}\) -symmetry(hereforth \(\partial _{\mathcal {P}\mathcal {T}}\) ) 被研究在典型的福克空间设置。所述 Fock 空间被理解为再生核希尔伯特空间 (RKHS)。对于非厄米 Bose-Hubbard 型哈密顿量（涉及两个玻色子算子）及其本征态，分析了相同类型的对称性。对称破缺现象也被考虑。