The horizons of observability in PT-symmetric four-site quantum lattices

The “Hermitizability” problem of quantum theory is explained, discussed and illustrated via the discrete-lattice cryptohermitian quantum graphs. In detail, the description of the domain $\mathcal{D}$ of admissible parameters is provided for the “circular-model” three-parametric quantum Hamiltonian $H$ using periodic boundary conditions. It is emphasized that even in such an elementary system the weak- and strong-coupling subdomains of $\mathcal{D}$ become, unexpectedly, non-empty and disconnected.