Classical Propagation in the Quantum Inverted Oscillator

We emphasize the fact the evolution of quantum states in the inverted oscillator (IO) is reduced to classical equations of motion, stressing that the corresponding tunnelling and reflexion coefficients addressed in the literature are calculated by considering only classically trajectories. The Wigner function formalism is employed to describe the IO classical dynamics, subsequently leading to the introduction of the Ambiguity function lying in the so-called Reciprocal phase space. Our findings, show that the Ambiguity function behavior, subjected to the IO, allude a classical propagation with an associated integral of motion, and complex conjugated doubly degenerate energy states.