  In the global framework of quantum theory the individual quantum systems seem
clearly separated into two families with the respective manifestly Hermitian
and hiddenly Hermitian operators of their Hamiltonian. In the light of certain
preliminary studies these two families seem to have an empty overlap. In this
paper we demonstrate that it is not so. We are going to show that whenever the
interaction potentials are chosen weakly nonlocal, the separation of the two
families may disappear. The overlaps alias interfaces between the Hermitian and
non-Hermitian descriptions of the unitarily evolving quantum system in question
may become non-empty. This assertion will be illustrated via a few analytically
solvable elementary models.
