Quasi-Hermitian quantum mechanics and a new class of user-friendly matrix Hamiltonians

• URL: http://arxiv.org/abs/2308.15043v1
• Date: Tue, 29 Aug 2023 05:54:43 GMT
• Title: Quasi-Hermitian quantum mechanics and a new class of user-friendly matrix Hamiltonians
• Authors: Olaf Lechtenfeld and Miloslav Znojil
• Abstract summary: We introduce a new family of Hamiltonians $H$ for which the metrics $Theta$ become available in closed form. Although acting in another, manifestly unphysical Hilbert space $cal H$, the amended Hamiltonian $H neq Hdagger$ can be perceived as self-adjoint.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In the conventional Schr\"{o}dinger's formulation of quantum mechanics the unitary evolution of a state $\psi$ is controlled, in Hilbert space ${\cal L}$, by a Hamiltonian $\mathfrak{h}$ which must be self-adjoint. In the recent, ``quasi-Hermitian'' reformulation of the theory one replaces $\mathfrak{h}$ by its isospectral but non-Hermitian avatar $H = \Omega^{-1}\mathfrak{h}\Omega$ with $\Omega^\dagger\Omega = \Theta \neq I$. Although acting in another, manifestly unphysical Hilbert space ${\cal H}$, the amended Hamiltonian $H \neq H^\dagger$ can be perceived as self-adjoint with respect to the amended inner-product metric $\Theta$. In our paper motivated by a generic technical ``user-unfriendliness'' of the non-Hermiticity of $H$ we introduce and describe a specific new family of Hamiltonians $H$ for which the metrics $\Theta$ become available in closed form.

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