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

Abstract

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

arXiv comment28 pp

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• Hybrid form of quantum theory with non-Hermitian Hamiltonians
• Systematics of quasi-Hermitian representations of non-Hermitian quantum models
• PT-symmetric dynamical confinement: Fermi acceleration, quantum force and Berry phase
• Factorized Hilbert-space metrics and non-commutative quasi-Hermitian observables
• Non-stationary quantum mechanics in hybrid non-Hermitian interaction representation