Feasibility and method of multi-step Hermitization of crypto-Hermitian quantum Hamiltonians

In the popular ${\cal PT}-$symmetry-based formulation of quantum mechanics of closed systems one can build unitary models using non-Hermitian Hamiltonians (i.e., $H \neq H^\dagger$) which are Hermitizable (so that one can write, simultaneously, $H = H^\ddagger$). The essence of the trick is that the reference Hilbert space $\cal R$ (in which we use the conventional inner product $\langle \psi_a|\psi_b\rangle$ and write $H \neq H^\dagger$) is declared unphysical. The necessary Hermiticity of the Hamiltonian $H = H^\ddagger$ can be then achieved by the mere metric-mediated amendment $\langle \psi_a|\Theta|\psi_b\rangle$ to the inner product. This converts $\cal R$ into a correct physical Hilbert space $\cal H$. The feasibility of the construction is based on a factorization postulate $\Theta={\cal PC}$ where, usually, ${\cal P}$ is parity and ${\cal C}$ is charge. In our paper we propose a more general factorization recipe in which one constructs $\Theta=Z_NZ_{N-1}\ldots Z_1$, at any $N$, in terms of suitable auxiliary pre-metric operators $Z_k$.