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Both real and complex semiclassical eigen energies of two dimensional non-Hermitian Hamiltonian systems are obtained by classical (Lie transform) perturbation theory requiring the action variables I 1 and I 2 to satisfy the quantization condition I 1 =( n 1 +(1/2)) ℏ and I 2 =( n 2 +(1/2)) ℏ respectively where n 1 , n 2 are integers. Classical perturbation theory with Lie transform makes classical trajectories, which are non-periodic or non-quasi-periodic, periodic. It was observed that this method produces accurate eigen energies even when classical trajectories are not periodic or quasi-periodic. Eigen energies obtained by classical perturbation theory are compared with the same, determined by Rayleigh-Schroedinger perturbation theory

Semiclassical quantization of non-Hermitian 2-D systems: Classical (Lie transform) Perturbation theory