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A $(K+1)-$plet of non-Hermitian and time-dependent operators (say,$\Lambda_j(t)$, $j=0,1,\ldots,K$) can be interpreted as the set of observablescharacterizing a unitary quantum system. What is required is the existence of aself-adjoint and, in general, time-dependent operator (say, $\Theta(t)$ calledinner product metric) making the operators quasi-Hermitian,$\Lambda_j^\dagger(t)\Theta(t)=\Theta(t)\Lambda_j(t)$. The theory (callednon-Hermitian interaction-picture, NIP) requires a separate description of theevolution of the states $\psi(t)$ (realized, via Schr\"{o}dinger-type equation,by a generator, say, $G(t)$) and of the observables themselves (a differentgenerator (say, $\Sigma(t)(t)$) occurs in the related non-HermitianHeisenberg-type equation). Every $\Lambda_j(t)$ (and, in particular,Hamiltonian $H(t)=\Lambda_0(t)$) appears isospectral to its hypotheticalisospectral and self-adjoint (but, by assumption, prohibitivelyuser-unfriendly) avatar $\lambda_j(t)=\Omega(t)\Lambda_j(t)\Omega^{-1}(t)$ with$\Omega^\dagger(t)\Omega(t)=\Theta(t)$. In our paper the key role played byidentity $H(t)=G(t)+\Sigma(t)$ is shown to imply that there exist just threealternative meaningful implementations of the NIP approach, viz., ``numberone'' (a ``dynamical'' strategy based on the knowledge of $H(t)$), ``numbertwo'' (a ``kinematical'' one, based on the Coriolis force $\Sigma(t)$) and``number three'' (in the literature, such a construction based on $G(t)$ ismost popular but, paradoxically, it is also most complicated).

Three alternative model-building strategies using quasi-Hermitian time-dependent observables