Short Time Dynamics of Scalar Products in Hilbert Space

We use the semiclassical method proposed in \cite{Adelcio2003} to study scalar products such as the overlap, Husimi functions and fidelity decay. Scars of classical periodic orbits arise naturally in this pertubative expansion. We also derive analytically a well known numerical result that fidelity has a quadratic decay for short times. We study the overlap in the chaotic regime and integrable for some simple systems.