On the invariant method for the time-dependent non-Hermitian Hamiltonians

. We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators H ( t ) that generate a real phase in their time evolution. This involves the use of invariant operators I P H ( t ) that are pseudo-Hermitian with respect to the time-dependent metric operator, which implies that the dynamics is governed by unitary time evolution. Furthermore, H ( t ) is generally not quasi-Hermitian and does not define an observable of the system but I P H ( t ) obeys a quasi-Hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.