  Let us consider the quantum/versus classical dynamics for Hamiltonians of the
form \beq \label{0.1} H\_{g}^{\epsilon} := \frac{P^2}{2}+ \epsilon
\frac{Q^2}{2}+ \frac{g^2}{Q^2} \edq where $\epsilon = \pm 1$, $g$ is a real
constant. We shall in particular study the Quantum Fidelity between
$H\_{g}^{\epsilon}$ and $H\_{0}^{\epsilon}$ defined as \beq \label{0.2}
F\_{Q}^{\epsilon}(t,g):= < \exp(-it H\_{0}^{\epsilon})\psi, exp(-itH\_{g}^
{\epsilon})\psi > \edq for some reference state $\psi$ in the domain of the
relevant operators. We shall also propose a definition of the Classical
Fidelity, already present in the literature (\cite{becave1}, \cite{becave2},
\cite{ec}, \cite{prozni}, \cite{vepro}) and compare it with the behaviour of
the Quantum Fidelity, as time evolves, and as the coupling constant $g$ is
varied.
