  Examples are given of non-Hermitian Hamiltonian operators which have a real
spectrum. Some of the investigated operators are expressed in terms of the
generators of the Weil-Heisenberg algebra. It is argued that the existence of
an involutive operator $\hat J$ which renders the Hamiltonian $\hat
J$-Hermitian leads to the unambiguous definition of an associated positive
definite norm allowing for the standard probabilistic interpretation of quantum
mechanics. Non-Hermitian extensions of the Poeschl-Teller Hamiltonian are also
considered. Hermitian counterparts obtained by similarity transformations are
constructed.
