By constructing a non-unitary but Hermitian transformation operator,we prove that the non-Hermitian Hamiltonian for a periodically driven harmonic oscillator is a pseudo-Hermitian Hamiltonian,characterized by real eigenvalues.For the time-dependent non-Hermitian Hamiltonian,both the metric and transformation operators are shown to be time-dependent.Analytic quantum wave functions of the corresponding Hermitian Hamiltonian are obtained from the dual Schrödinger equations respectively for the"bra"and"ket"states.Moreover,the classical correspondence of the non-Hermitian Hamiltonian is revealed through classical canonical transformations.The relation between quantum LR phase and classical angle is found in one period of the driving field.By analyzing the non-Hermitian Hamiltonian of periodic driven harmonic oscillator,it is concluded while a Hermitian Hamiltonian is sufficient for having real eigenvalues,it is not a necessary condition.The Hermitian counterpart of the non-Hermitian Hamiltonian can be obtained by a generalized gauge transformation that utilizes a non-unitary,but Hermitian transformation operator.The results of this work pave the way for new explorations into the non-Hermitian Hamiltonian system.
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