Quantum and classical fidelity for singular perturbations of the inverted and harmonic oscillator

Let us consider the quantum/versus classical dynamics for Hamiltonians of the formequation(0.1)Hgϵ:=P22+ϵQ22+g2Q2, where ϵ=±1ϵ=±1, g   is a real constant. We shall in particular study the quantum fidelity (Q.F.) between Hgϵ and H0ϵ defined asequation(0.2)FQϵ(t,g):=〈exp(−itH0ϵ)ψ,exp(−itHgϵ)ψ〉 for some reference state ψ in the domain of the relevant operators. We shall also propose a definition of the classical fidelity (C.F.), already present in the literature [G. Benenti, G. Casati, G. Veble, On the stability of classical chaotic motion under systems' perturbations, Phys. Rev. E 67 (2003) 055202(R); B. Eckhardt, Echoes in classical dynamical systems, J. Phys. A: Math. Gen. 36 (2003) 371–380; T. Prosen, M. Znidaric, Stability of quantum motion and correlation decay, J. Math. Phys. A: Math. Gen. 35 (2002) 1455–1481; G. Veble, T. Prosen, Faster than Lyapunov decays of classical Loschmidt Echo, Phys. Rev. Lett. 92 (2003) 034101] and compare it with the behavior of the quantum fidelity, as time evolves, and as the coupling constant g is varied.