A Real Expectation Value of the Time-dependent Non-Hermitian Hamiltonians* *This paper is dedicated to the memory of our friends and colleagues Lahcene Alouani and Mebrouk Gherda died due to covid 19.

Abstract

With the aim to solve the time-dependent Schrödinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent -symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the −inner product, where is a obtained from the charge conjugation operator through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the 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.