A real expectation value of the time-dependent non-Hermitian Hamiltonians

Abstract: With the aim to solve the time-dependent Schr\"{o}dinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent $\mathcal{PT}$-symmetric one. Consequently, the solution of time-dependent Schr\"{o}dinger equation becomes easily deduced and the evolution preserves the $\mathcal{C(}t\mathcal{)PT}$-inner product, where $\mathcal{C(}t\mathcal{)}$ is a obtained from the charge conjugation operator $\mathcal{C}$ through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the $\mathcal{C(}t\mathcal{)PT}$ normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.