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 IPH(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 IPH(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.