Hybrid form of quantum theory with non-Hermitian Hamiltonians

• URL: http://arxiv.org/abs/2211.10633v1
• Date: Sat, 19 Nov 2022 09:51:29 GMT
• Title: Hybrid form of quantum theory with non-Hermitian Hamiltonians
• Authors: Miloslav Znojil
• Abstract summary: Freeman Dyson discovered that the practical solution of Schr"odinger equation may be facilitated when one replaces the standard self-adjoint Hamiltonian $mathfrakh=mathfrakhdagger$. In our present letter we describe an alternative Hermitization of a hybrid form'' which simply combines the OT model-building strategy with the MA model-building strategy.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In 1956, Freeman Dyson discovered that the practical solution of Schr\"{o}dinger equation may be facilitated when one replaces the standard self-adjoint Hamiltonian $\mathfrak{h}=\mathfrak{h}^\dagger$ by its manifestly non-Hermitian isospectral avatar $H =\Omega^{-1} \mathfrak{h} \Omega$ with $\Omega^\dagger\Omega=\Theta \neq I$. The standard unitary interpretation of the evolution can be then achieved either by the ``operator transformation'' (OT) reconstruction of $\mathfrak{h}$, or via a metric-amendment (MA) change of the Hilbert space, ${\cal H}_{auxiliary}\to {\cal H}_{physical}$. In our present letter we describe an alternative Hermitization of a ``hybrid form'' (HF) which simply combines the OT model-building strategy with the MA model-building strategy. The merits of the approach are illustrated using a schematic two-state quantum system.

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