Hybrid form of quantum theory with non-Hermitian Hamiltonians

In 1956, Freeman Dyson discovered that the practical solution of Schrödinger equation may be facilitated when one replaces the standard self-adjoint Hamiltonian $\mathfrak{h}={\mathfrak{h}}^{†}$ by its manifestly non-Hermitian isospectral avatar $H={\mathrm{\Omega }}^{-1}\mathfrak{h}\mathrm{\Omega }$ with ${\mathrm{\Omega }}^{†}\mathrm{\Omega }=\mathrm{\Theta }\mathrm{\ne }I$. The standard unitary interpretation of the evolution can be then achieved either by the operator transformation'' (OT) reconstruction of $\mathfrak{h}$, or via a metric-amendment (MA) change of the Hilbert space, ${\cal H}_{auxiliary}\to {\cal H}_{physical}$. In our present letter we describe an alternative Hermitization of a hybrid form’’ (HF) which simply combines the OT model-building strategy with the MA model-building strategy. The merits of the approach are illustrated using a schematic two-state quantum system.