Quantum and Classical Fidelity for Singular Perturbations of the Inverted and Harmonic Oscillator

Abstract

Let us consider the quantum/versus classical dynamics for Hamiltonians of the form \beq \label{0.1} H\_{g}^ε := \frac{P^2}{2}+ ε\frac{Q^2}{2}+ \frac{g^2}{Q^2} \edq where $ε= \pm 1$, $g$ is a real constant. We shall in particular study the Quantum Fidelity between $H\_{g}^ε$ and $H\_{0}^ε$ defined as \beq \label{0.2} F\_{Q}^ε(t,g):= < \exp(-it H\_{0}^ε)ψ, exp(-itH\_{g}^ ε)ψ> \edq for some reference state $ψ$ 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.