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

Abstract

In the framework of the so-called quasi-Hermitian quantum mechanics of stationary unitary systems, bound states are usually constructed as eigenstates $|{\psi }_n⟩$ of a Hamiltonian operator $H$ with real spectrum which is non-Hermitian, $H\ne H^{†}$. One of the ways of the standard probabilistic interpretation of such systems consists in a transformation of $H$ into its isospectral Hermitian ``twin" $\mathfrak{h}={\mathfrak{h}}^{†}$ via one of the so-called Dyson maps $\Omega :H\to \mathfrak{h}$. Naturally, the well known ambiguity of these $H-$dependent Dyson-map transformations implies also an ambiguity of the physical, $\Omega -$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 $\Omega =\Omega (H)$ 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 $H$.

arXiv comment21 pp., written in honor of our friend, professor Jean-Pierre Gazeau, on the occasion of his 80th birthday

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