Twin Hamiltonians, three types of the Dyson maps, and the probabilistic interpretation problem in quasi-Hermitian quantum mechanics

• URL: http://arxiv.org/abs/2511.20404v1
• Date: Tue, 25 Nov 2025 15:33:31 GMT
• Title: Twin Hamiltonians, three types of the Dyson maps, and the probabilistic interpretation problem in quasi-Hermitian quantum mechanics
• Authors: Aritra Ghosh, Adam Miranowicz, Miloslav Znojil,
• Abstract summary: In quasi-Hermitian quantum mechanics, an optimal, calculation-friendly form of Hamiltonian is generally non-Hermitian, $H neq Hdagger$.<n>Here, we focus on an alternative strategy: transforming $H$ into a Hermitian form via the Dyson map $: H to mathfrakh$.<n>This construction of the Hermitian isospectral twin $mathfrakh$ of $H$ does not only restore the conventional correspondence principle between quantum and classical physics, but it also provides a framework for the exhaustive
• Score: 3.7723788828505125
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In quasi-Hermitian quantum mechanics (QHQM) of unitary systems, an optimal, calculation-friendly form of Hamiltonian is generally non-Hermitian, $H \neq H^\dagger$. This makes its physical interpretation ambiguous. Without altering $H$, this ambiguity is resolved by specifying a nontrivial inner-product metric $Θ$ in Hilbert space. Here, we focus on an alternative strategy: transforming $H$ into a Hermitian form via the Dyson map $Ω: H \to \mathfrak{h}$. This construction of the Hermitian isospectral twin $\mathfrak{h}$ of $H$ does not only restore the conventional correspondence principle between quantum and classical physics, but it also provides a framework for the exhaustive classification of all admissible probabilistic interpretations of quantum systems in QHQM framework.

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