Systematics of quasi-Hermitian representations of non-Hermitian quantum models

• URL: http://arxiv.org/abs/2212.03940v1
• Date: Wed, 7 Dec 2022 20:10:58 GMT
• Title: Systematics of quasi-Hermitian representations of non-Hermitian quantum models
• Authors: Miloslav Znojil
• Abstract summary: This paper introduces and describes a set of constructive returns of the description to one of the correct and eligible physical Hilbert spaces $cal R_N(0)$. In the extreme of the theory the construction is currently well known and involves solely the inner product metric $Theta=Theta(H)$. At $j=N$ the inner-product metric remains trivial and only the Hamiltonian must be Hermitized, $H to mathfrakh = Omega,H,Omega-1=mathfrak
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In the recently quickly developing context of quantum mechanics of unitary systems using a time-independent non-Hermitian Hamiltonian $H$ (having real spectrum and defined as acting in an unphysical but user-friendly Hilbert space ${\cal R}_N^{(0)}$), the present paper introduces and describes a set of constructive returns of the description to one of the correct and eligible physical Hilbert spaces ${\cal R}_0^{(j)}$. The superscript $j$ may run from $j=0$ to $j=N$. In the $j=0$ extreme of the theory the construction is currently well known and involves solely the inner product metric $\Theta=\Theta(H)$. The Hamiltonian $H$ itself remains unchanged. At $j=N$ the inner-product metric remains trivial and only the Hamiltonian must be Hermitized, $H \to \mathfrak{h} = \Omega\,H\,\Omega^{-1}=\mathfrak{h}^\dagger$. At the remaining superscripts $j=1,2,\ldots, N-1$, a new, hybrid form of the construction of a consistent quantum model is proposed, requiring a simultaneous amendment of both the metric and the Hamiltonian. In applications, one of these options is expected to be optimal for a given $H$ in a way illustrated by a schematic three-state example.

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