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

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Let us consider the quantum/versus classical dynamics for Hamiltonians of the form 0.1 H_gε:= P²/2+ εQ²/2+ g²/Q² where ε= ± 1, g is a real constant. We shall in particular study the Quantum Fidelity between H_gε and H_0ε defined as 0.2 F_Qε(t,g):= < (-it H_0ε)ψ, exp(-itH_g ε)ψ> 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 (becave1, becave2, ec, prozni, vepro) and compare it with the behaviour of the Quantum Fidelity, as time evolves, and as the coupling constant g is varied.