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An arbitrary form of complex potential perturbation in an oscillator consistsof many exciting questions in open quantum systems. These often providevaluable insights in a realistic scenario when a quantum system interacts withexternal environments. Action renormalization will capture the phase of thewave functions; hence we construct wave function from Bethe ansatz andFrobenius methods. The unitary and non-unitary regimes are discussed to connectwith functional calculations. We present a functional renormalizationcalculation for a non-hermitian oscillator. A dual space Left-Right formulationis worked out in functional bosonic variables to derive the flow equation forscale dependent action. We show equivalence between vertex operator andpermutation operators. The results can be compared with Wentzel KramersBrillouin(WKB) calculation. We formally construct the Bosonic coherent statesin the dual space;breaking symmetry will lead to anyonic coherent states. Thelimit cycle in renormalization trajectories for complex flow parameters,especially in extended, complex time limits indicates the need for revisitingthe Bendixson theorem.

Functional Renormalization analysis of Bose-Einstien Condensation  through complex interaction in Harmonic Oscillator; Can Bendixson criteria be  extended to complex time?