Invariant operators and Heisenberg operators for time-dependent generalized oscillators

We investigate the relation between the invariant operators satisfying the quantum Liouville equation and the Heisenberg operators satisfying the Heisenberg equation. For time-dependent generalized oscillators, we find the invariant operators, also known as the Ermakov-Lewis invariants, le in terms of a complex classical solution. We also find the evolution operator and the Heisenberg position and momentum operators in terms of the complex classical solution. Physical quantities, such as correlation functions, are calculated using both the invariant operators and the Heisenberg operators.