Abstract
In an amended version of non-Hermitian interaction picture we propose to work with the states ψ(t) in a dyadic representation. The control of evolution via two conjugate Schr\"{o}diner equations then renders the usual necessity of the construction of the time-dependent inner-product-metric operator Θ(t) redundant. The primary information about dynamics is assumed carried by a non-Hamiltonian observable (say, R(t)). A specific realization of phase transitions is then rendered possible via the Kato's exceptional-point (EP) degeneracy of the eigenvalues of R(t) at the EP time t=t^(EP). For illustration a cosmological model is proposed mimicking the unitary-evolution birth of the Universe from an initial quantum Big Bang singularity.