  In this work we intend to study a class of time-dependent quantum systems
with non-Hermitian Hamiltonians, particularly those whose Hermitian counterpart
are important for the comprehension of posed problems in quantum optics and
quantum chemistry, which consists of an oscillator with time-dependent mass and
frequency under the action of a time-dependent imaginary potential. The
propagator for a general time-dependence of the parameters and the
wave-functions are obtained explicitly for constant frequency and mass and a
linear time-dependence in the potential. The wave-functions are used to obtain
the expectation value of the Hamiltonian. Although it is neither Hermitian nor
PT symmetric, the case under study exhibits real values of energy.
