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Crypto-unitary forms of quantum evolution operators

For the description of quantum evolution, the use of a manifestly time-dependent quantum Hamiltonian $\mathfrak{h}(t) =\mathfrak{h}^\dagger(t)$ is shown equivalent to the work with its simplified, time-independent alternative $G\neq G^\dagger$. A tradeoff analysis is performed recommending the latter option. The physical unitarity requirement is shown fulfilled in a suitable ad hoc representation of Hilbert space.