What question did this study set out to answer?

The aim is to construct isospectral Hermitian forms of non-Hermitian Hamiltonians in quasi-Hermitian quantum mechanics.

January 22, 2026Open Access

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

Key Points

The aim is to construct isospectral Hermitian forms of non-Hermitian Hamiltonians in quasi-Hermitian quantum mechanics.
Transformation of Hamiltonian H into its isospectral Hermitian form using Dyson map Ω.
Classification of admissible probabilistic interpretations by selecting Hamiltonians as per quasi-Hermitian framework.
Systematic construction of Hermitian twins h based on transformations of H.
Establishment of a complete set of Hermitian isospectral twins h associated with any given non-Hermitian Hamiltonian H.
Restoration of the correspondence principle between quantum and classical mechanics.
Development of a framework for classifying probabilistic interpretations in quasi-Hermitian quantum systems.

Abstract

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.

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