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 formwhere = ±1, g is a real constant. We shall in particular study the quantum fidelity (Q.F.) between H g and H 0 defined asfor 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.

Coherent states of the inverted Caldirola–Kanai oscillator with time-dependent singularities

“…A generalization of Calogero-Sutherland model is achieved by Park, with a condition that the strength of the inverse quadratic potential is not fixed but dynamical bounded by below [30]. Recently, the quantum properties such as quantum fidelity for the inverted oscillator perturbed by an inverse quadratic potential are investigated in the literature [31].…”

The quantum fidelity for the time-periodic singular harmonic oscillator

“…Thus it would be desirable to understand more deeply the relationships between the Classical and Quantum Fidelity on a firm mathematical basis. We shall pursue this investigation in future publications, notably in the semiclassical limit (see [10], [8]).…”