Twin Hamiltonians, three types of the Dyson maps, and the probabilistic interpretation problem in quasi-Hermitian quantum mechanics
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| Format: | Preprint |
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2025
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| _version_ | 1866912832983924736 |
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| author | Ghosh, Aritra Miranowicz, Adam Znojil, Miloslav |
| author_facet | Ghosh, Aritra Miranowicz, Adam Znojil, Miloslav |
| contents | In the framework of the so-called quasi-Hermitian quantum mechanics of stationary unitary systems, bound states are usually constructed as eigenstates $|ψ_n \rangle$ of a Hamiltonian operator $H$ with real spectrum which is non-Hermitian, $H \neq H^\dagger$. One of the ways of the standard probabilistic interpretation of such systems consists in a transformation of $H$ into its isospectral Hermitian ``twin" $\mathfrak{h}= \mathfrak{h}^\dagger$ via one of the so-called Dyson maps $Ω: H \to \mathfrak{h}$. Naturally, the well known ambiguity of these $H-$dependent Dyson-map transformations implies also an ambiguity of the physical, $Ω-$dependent probabilistic and experimental interpretation of the system in question. In the present paper, an exhaustive classification of all of the eligible $H-$dependent Dyson maps $Ω=Ω(H)$ is provided, implying also a systematic framework for a specification of all of the possible probabilistic interpretations of the quantum system characterized by a preselected $H$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_20404 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Twin Hamiltonians, three types of the Dyson maps, and the probabilistic interpretation problem in quasi-Hermitian quantum mechanics Ghosh, Aritra Miranowicz, Adam Znojil, Miloslav Quantum Physics In the framework of the so-called quasi-Hermitian quantum mechanics of stationary unitary systems, bound states are usually constructed as eigenstates $|ψ_n \rangle$ of a Hamiltonian operator $H$ with real spectrum which is non-Hermitian, $H \neq H^\dagger$. One of the ways of the standard probabilistic interpretation of such systems consists in a transformation of $H$ into its isospectral Hermitian ``twin" $\mathfrak{h}= \mathfrak{h}^\dagger$ via one of the so-called Dyson maps $Ω: H \to \mathfrak{h}$. Naturally, the well known ambiguity of these $H-$dependent Dyson-map transformations implies also an ambiguity of the physical, $Ω-$dependent probabilistic and experimental interpretation of the system in question. In the present paper, an exhaustive classification of all of the eligible $H-$dependent Dyson maps $Ω=Ω(H)$ is provided, implying also a systematic framework for a specification of all of the possible probabilistic interpretations of the quantum system characterized by a preselected $H$. |
| title | Twin Hamiltonians, three types of the Dyson maps, and the probabilistic interpretation problem in quasi-Hermitian quantum mechanics |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2511.20404 |