Understanding Partial $\mathcal {P}\mathcal {T}$ Symmetry as Weighted Composition Conjugation in Reproducing Kernel Hilbert Space: An application to Non-hermitian Bose-Hubbard Type Hamiltonian in Fock space

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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.