Generalized Gauge Transformation with PT$PT$‐Symmetric Non‐Unitary Operator and Classical Correspondence of Non‐Hermitian Hamiltonian for a Periodically Driven System

We in this paper demonstrate that the $PT$-symmetric non-Hermitian Hamiltonian for a periodically driven system can be generated from kernel by generalized gauge transformation. The is Hermitian and static, while time-dependent transformation operator has to $PT$ symmetric non-unitary general. Biorthogonal sets of eigenstates appear necessarily as consequence Hamiltonian. obtain analytically wave functions associated non-adiabatic Berry phase $\gamma_{n}$ $n$th eigenstate. classical version becomes complex function canonical variables time. corresponding derived with canonical-variable transfer Moreover, change position-momentum angle-action it revealed Hannay's angle $\Delta \theta_{H}$ satisfy precisely quantum-classical correspondence,$\gamma_{n}=$ $(n+1/2)\Delta \theta_{H}$.