The propagator of the inverted Caldirola-Kanai Oscillator

In this paper, we will derive the propagators for an inverted Caldirola-Kanai oscillator by the Feynman path integral, the Schwinger method, and the Dodonov method. In Feynman path inte-gral, the propagator can be calculated from the functional integrals while in the Schwinger method the propagator can be obtained from basic operator algebra and elementary integrations. In Dodonov method, the propagator can be derived from the application of the integrals of the motion of quantum systems. Furthermore, the connection between these methods is also discussed.