Twin Hamiltonians, Alternative Parametrizations of the Dyson Maps, and the Probabilistic Interpretation Problem in Quasi-Hermitian Quantum Mechanics

In quasi-Hermitian quantum mechanics (QHQM) of unitary systems, an optimal, calculation-friendly form of Hamiltonian is generally non-Hermitian, H≠H†. This makes its physical interpretation ambiguous. Without altering H, this ambiguity can be resolved either via a transformation of H into its isospectral Hermitian form via a so-called Dyson map Ω:H→h, or via a (formally equivalent) specification of a nontrivial physical inner-product metric Θ in Hilbert space. Here, we focus on the former strategy. Our present construction of the Hermitian isospectral twins h of H is exhaustive. As a byproduct, it not only restores the conventional correspondence principle between quantum and classical physics, but it also provides a framework for a systematic classification of all of the admissible probabilistic interpretations of quantum systems using a preselected H in QHQM framework.